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All questions of Geometry for CTET & State TET Exam
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The angles of a quadrilateral are in the ratio 4: 5: 10: 11. The angles are- a)36°, 60°, 108°, 156°
- b)52°, 60°, 122°, 126°
- c)48°, 60°, 120°, 132°
- d)60°, 60°, 120°, 120°
Correct answer is option 'C'. Can you explain this answer?
The angles of a quadrilateral are in the ratio 4: 5: 10: 11. The angles are
a)
36°, 60°, 108°, 156°
b)
52°, 60°, 122°, 126°
c)
48°, 60°, 120°, 132°
d)
60°, 60°, 120°, 120°
|
Raghav Bansal answered |
Let x be the common angle among all the four angles of a quadrilateral.
As per angle sum property, we know:
4x+5x+10x+11x = 360°
30x = 360°
x = 12°
Hence, angles are
4x = 4 (12) = 48°
5x = 5 (12) = 60°
10x = 10 (12) = 120°
11x = 11 (12) = 132°
A rectangular enclosure 40 m x 36 m has a horse tethered to a corner with a rope of 14 m in length. What is the ratio of the respective areas it can graze, if it is outside the enclosure and if it is inside the enclosure?- a)2
- b)2:3
- c)1:4
- d)3:1
Correct answer is option 'D'. Can you explain this answer?
A rectangular enclosure 40 m x 36 m has a horse tethered to a corner with a rope of 14 m in length. What is the ratio of the respective areas it can graze, if it is outside the enclosure and if it is inside the enclosure?
a)
2
b)
2:3
c)
1:4
d)
3:1
|
Arun Sharma answered |
- Imagine a circle on the corner of the rectangle.
- 3 quarters of the circle lie outside the rectangle and 1 quarter lies inside.
- Hence, Required ratio = 3 : 1
If PQR is an isosceles triangle and M is a point on QR such that PM⊥QR,then- a)PQ2−PM2 = QM2−MR2.
- b)PQ2+PM2 = QM.MR.
- c)PQ2−PR2 = QM2 − MR2
- d)PQ2−PM2 = QM.MR.
Correct answer is option 'C'. Can you explain this answer?
If PQR is an isosceles triangle and M is a point on QR such that PM⊥QR,then
a)
PQ2−PM2 = QM2−MR2.
b)
PQ2+PM2 = QM.MR.
c)
PQ2−PR2 = QM2 − MR2
d)
PQ2−PM2 = QM.MR.
|
Krishna Iyer answered |
Since, in a triangle the sum of squares of any wo sides is equal to twice the square of half of the third side together with twice the square of the median bisecting it.
In ΔPQM,
In ΔPQM,
In the adjoining figure D, E and F are the mid-points of the sides BC, AC and AB respectively. ΔDEF is congruent to triangle :
- a)ABC
- b)AEF
- c)AFE, BFD and CDE
- d)CDE , BFD
Correct answer is option 'C'. Can you explain this answer?
In the adjoining figure D, E and F are the mid-points of the sides BC, AC and AB respectively. ΔDEF is congruent to triangle :
a)
ABC
b)
AEF
c)
AFE, BFD and CDE
d)
CDE , BFD
|
|
Raghav Bansal answered |
Given :△ABC, F, D and E are mid points of AB, BC, CA respectively.
Using mid point theorem we prove that □AEFD, □DBFF and □DCEF are parallelograms. The diagonal of a parallelogram divides the parallelogram into two congruent triangles. So all triangles are congruent to each other. So ΔDEF is congruent to
△AFE, △BFD and △CDE
Using mid point theorem we prove that □AEFD, □DBFF and □DCEF are parallelograms. The diagonal of a parallelogram divides the parallelogram into two congruent triangles. So all triangles are congruent to each other. So ΔDEF is congruent to
△AFE, △BFD and △CDE
- a)∠B=∠D.
- b)∠B=∠E.
- c)∠A=∠D.
- d)∠A=∠F.
Correct answer is option 'A'. Can you explain this answer?
a)
∠B=∠D.
b)
∠B=∠E.
c)
∠A=∠D.
d)
∠A=∠F.
|
Manisha reddy answered |
Given information:
- QR < 2pq="" -="" />
- PR = PQ + 10
- Perimeter = 40
To find:
- Smallest side of the triangle PQR
Solution:
Let's assume that PQ = x, QR = y, and PR = z.
Using the given information, we can write two equations based on the lengths of the sides:
1. y = 2x - 2 (QR is less than twice the length of PQ by 2 cm)
2. z = x + 10 (PR exceeds the length of PQ by 10 cm)
We also know that the perimeter of the triangle is 40 cm:
x + y + z = 40
Substituting the values of y and z from equations (1) and (2) into equation (3), we get:
x + (2x - 2) + (x + 10) = 40
Simplifying the equation, we get:
4x + 8 = 40
4x = 32
x = 8
Therefore, the length of the smallest side of the triangle PQR is PQ = x = 8 cm.
Answer: Option (B) 8 cm.
- QR < 2pq="" -="" />
- PR = PQ + 10
- Perimeter = 40
To find:
- Smallest side of the triangle PQR
Solution:
Let's assume that PQ = x, QR = y, and PR = z.
Using the given information, we can write two equations based on the lengths of the sides:
1. y = 2x - 2 (QR is less than twice the length of PQ by 2 cm)
2. z = x + 10 (PR exceeds the length of PQ by 10 cm)
We also know that the perimeter of the triangle is 40 cm:
x + y + z = 40
Substituting the values of y and z from equations (1) and (2) into equation (3), we get:
x + (2x - 2) + (x + 10) = 40
Simplifying the equation, we get:
4x + 8 = 40
4x = 32
x = 8
Therefore, the length of the smallest side of the triangle PQR is PQ = x = 8 cm.
Answer: Option (B) 8 cm.
If bisectors of ∠A and ∠B of a quadrilateral ABCD intersect each other at P, of ∠B and ∠C at Q, of ∠C and ∠D at R and of ∠D and ∠A at S, then PQRS is a- a)Rectangle
- b)Quadrilateral whose opposite angles are supplementary
- c)Parallelogram
- d)Rhombus
Correct answer is option 'B'. Can you explain this answer?
If bisectors of ∠A and ∠B of a quadrilateral ABCD intersect each other at P, of ∠B and ∠C at Q, of ∠C and ∠D at R and of ∠D and ∠A at S, then PQRS is a
a)
Rectangle
b)
Quadrilateral whose opposite angles are supplementary
c)
Parallelogram
d)
Rhombus
|
|
Ananya Das answered |
To show: ∠PSR + ∠PQR = 180°
∠SPQ + ∠SRQ = 180°
∠SPQ + ∠SRQ = 180°
In △DSA,
∠DAS + ∠ADS + ∠DSA = 180° (angle sum property)
+ ∠ SA = 180° (since RD and AP are bisectors of ∠D and ∠A)
∠DSA = 180°
∠PSR = 180°−
∠PSR = 180°−
(∵ ∠DSA = ∠PSR are vertically opposite angles)
Similarly,
∠PQR = 180°− 
Adding (i) and (ii), we get, ∠PSR + ∠PQR = 180°
=360° − 1/2 × (∠A + ∠B + ∠C + ∠D)
=360°− 1/2 × 360° = 180° ∴ ∠PSR + ∠PQR = 180°
In quadrilateral PQRS,
∠SPQ + ∠SRQ + ∠PSR + ∠PQR = 360°
=> ∠SPQ + ∠SRQ + 180° = 360°
=> ∠SPQ + ∠SRQ = 180°
=> ∠SPQ + ∠SRQ + 180° = 360°
=> ∠SPQ + ∠SRQ = 180°
Hence, showed that opposite angles of PQRS are supplementary.
The areas of two similar triangles are 100cm2 and 49 cm2. If the altitude of the larger triangle is 5 cm, then the corresponding altitude of the other triangle is equal to- a)3.5 cm.
- b)3.9 cm.
- c)5.4 cm.
- d)4.5 cm.
Correct answer is option 'A'. Can you explain this answer?
The areas of two similar triangles are 100cm2 and 49 cm2. If the altitude of the larger triangle is 5 cm, then the corresponding altitude of the other triangle is equal to
a)
3.5 cm.
b)
3.9 cm.
c)
5.4 cm.
d)
4.5 cm.
|
|
Krishna Iyer answered |
Let the two similar triangles be ΔABC and ΔDEF such that ar(ΔABC) = 100 cm2 and ar(ΔDEF) = 49 cm2.
Let AM and DN be the respective altitudes of ΔABC and ΔDEF.
Since the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding altitudes
Let AM and DN be the respective altitudes of ΔABC and ΔDEF.
Since the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding altitudes
The line segments joining the mid points of the sides of a triangle form four triangles each of which is :- a)Similar to the original triangle
- b)Congruent to the original triangle
- c)An equilateral triangle
- d)An isosceles triangle
Correct answer is option 'A'. Can you explain this answer?
The line segments joining the mid points of the sides of a triangle form four triangles each of which is :
a)
Similar to the original triangle
b)
Congruent to the original triangle
c)
An equilateral triangle
d)
An isosceles triangle
|
|
Krishna Iyer answered |
Given :△ABC, D, E and F are mid points of AB, BC, CA respectively.
Using mid point theorem we prove that □ADEF, □DBEF and □DECF are parallelograms. The diagonal of a parallelogram divides the parallelogram into two congruent triangles. So all triangles are congruent to each other. And each small triangle is similar to the original triangle.
Using mid point theorem we prove that □ADEF, □DBEF and □DECF are parallelograms. The diagonal of a parallelogram divides the parallelogram into two congruent triangles. So all triangles are congruent to each other. And each small triangle is similar to the original triangle.
In a Quadrilateral ABCD, AB = BC and CD = DA, then the quadrilateral is a- a)Triangle
- b)Kite
- c)Rhombus
- d)Rectangle
Correct answer is option 'B'. Can you explain this answer?
In a Quadrilateral ABCD, AB = BC and CD = DA, then the quadrilateral is a
a)
Triangle
b)
Kite
c)
Rhombus
d)
Rectangle
|
|
Aradhiya Gupta answered |
All sides are equal in rhombus and square
In ΔABC, AB = 3 cm, AC = 4 cm and AD is the bisector of ∠A. Then, BD : DC is :
- a)9 : 16
- b)16 : 9
- c)3 : 4
- d)4 : 3
Correct answer is option 'C'. Can you explain this answer?
In ΔABC, AB = 3 cm, AC = 4 cm and AD is the bisector of ∠A. Then, BD : DC is :
a)
9 : 16
b)
16 : 9
c)
3 : 4
d)
4 : 3
|
Amrutha Banerjee answered |
Recent years, there has been an increasing focus on mental health and the importance of taking care of it. This is due to a growing recognition of the impact mental health has on overall well-being and the understanding that mental health issues are common and can affect anyone.
One reason for the increased focus on mental health is the rising prevalence of mental health disorders. According to the World Health Organization, one in four people in the world will be affected by mental or neurological disorders at some point in their lives. This means that mental health issues are widespread and can have a significant impact on individuals, families, and communities.
Another reason for the increased focus on mental health is the growing understanding of the link between mental health and physical health. Research has shown that mental health issues can lead to a range of physical health problems, including heart disease, diabetes, and obesity. Conversely, taking care of mental health can improve physical health outcomes and overall well-being.
Additionally, there has been a greater recognition of the stigma surrounding mental health and a push for more open discussions and conversations. People are becoming more willing to share their experiences with mental health and seek help when needed. This increased openness has contributed to a greater understanding and acceptance of mental health issues, leading to more resources and support being available.
In recent years, there has also been a growing emphasis on mental health in the workplace. Employers are recognizing the importance of supporting their employees' mental well-being and implementing measures to promote mental health in the workplace. This includes initiatives such as employee assistance programs, mental health training for managers, and creating a supportive work environment.
Overall, the increased focus on mental health in recent years is a positive development. It has led to greater awareness, understanding, and support for mental health issues. However, there is still work to be done to ensure that everyone has access to the resources and support they need for their mental well-being.
One reason for the increased focus on mental health is the rising prevalence of mental health disorders. According to the World Health Organization, one in four people in the world will be affected by mental or neurological disorders at some point in their lives. This means that mental health issues are widespread and can have a significant impact on individuals, families, and communities.
Another reason for the increased focus on mental health is the growing understanding of the link between mental health and physical health. Research has shown that mental health issues can lead to a range of physical health problems, including heart disease, diabetes, and obesity. Conversely, taking care of mental health can improve physical health outcomes and overall well-being.
Additionally, there has been a greater recognition of the stigma surrounding mental health and a push for more open discussions and conversations. People are becoming more willing to share their experiences with mental health and seek help when needed. This increased openness has contributed to a greater understanding and acceptance of mental health issues, leading to more resources and support being available.
In recent years, there has also been a growing emphasis on mental health in the workplace. Employers are recognizing the importance of supporting their employees' mental well-being and implementing measures to promote mental health in the workplace. This includes initiatives such as employee assistance programs, mental health training for managers, and creating a supportive work environment.
Overall, the increased focus on mental health in recent years is a positive development. It has led to greater awareness, understanding, and support for mental health issues. However, there is still work to be done to ensure that everyone has access to the resources and support they need for their mental well-being.
A semicircle is drawn on AC. Two chords AB and BC of length 8 cm and 6 cm respectively are drawn in the semicircle. What is the measure of the diameter of the circle?- a)14 cm.
- b)10 cm.
- c)12 cm.
- d)11 cm.
Correct answer is option 'B'. Can you explain this answer?
A semicircle is drawn on AC. Two chords AB and BC of length 8 cm and 6 cm respectively are drawn in the semicircle. What is the measure of the diameter of the circle?
a)
14 cm.
b)
10 cm.
c)
12 cm.
d)
11 cm.
|
|
Krishna Iyer answered |
The diameter of circle is AC. Here ∠ABC is angle of semicircle.
AB ∴ ∠ ABC = 90° ∴ Δ ABC is aright angled triangle.
10 cm
AB ∴ ∠ ABC = 90° ∴ Δ ABC is aright angled triangle.
10 cmIf area of a Parallelogram with sides ‘a’ and ‘b’ is A and that of a rectangle with sides ‘a’ and ‘b’ is B, then- a)A < B
- b)A = B
- c)A > B
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
If area of a Parallelogram with sides ‘a’ and ‘b’ is A and that of a rectangle with sides ‘a’ and ‘b’ is B, then
a)
A < B
b)
A = B
c)
A > B
d)
None of these
|
Man Oj answered |
Area of llgm=1/2 ab
area of rectangle=ab
since ab>1/2ab
A
area of rectangle=ab
since ab>1/2ab
A
ABCD is a Rectangle. Find the values of x and y?
AB =30 DA= 14 DC= x+y CB=x-y
- a)20 and 10
- b)25 and 5
- c)24 and 6
- d)22 and 8
Correct answer is option 'D'. Can you explain this answer?
ABCD is a Rectangle. Find the values of x and y?
AB =30 DA= 14 DC= x+y CB=x-y
a)
20 and 10
b)
25 and 5
c)
24 and 6
d)
22 and 8
|
Om Chawla answered |
ABCD is a rectangle.
∴ AB = CD
⇒ 30 = x + y
or x + y = 30 ..... (i)
Similarly, AD = BC
⇒ 14 = x - y
or x - y = 14 .......(ii)
On adding eq. (i) and (ii), we get
2x = 44
⇒ x = 22
Putting the value of x in eq. (i), we get
22 + y = 30
⇒ y = 30 -22
⇒ y = 8
So, x = 22, y = 8.
∴ AB = CD
⇒ 30 = x + y
or x + y = 30 ..... (i)
Similarly, AD = BC
⇒ 14 = x - y
or x - y = 14 .......(ii)
On adding eq. (i) and (ii), we get
2x = 44
⇒ x = 22
Putting the value of x in eq. (i), we get
22 + y = 30
⇒ y = 30 -22
⇒ y = 8
So, x = 22, y = 8.
ABCD is a Rhombus. Then, find the value of x and y?- a)40 and 40
- b)35 and 35
- c)37 and 37
- d)45 and 45
Correct answer is option 'B'. Can you explain this answer?
ABCD is a Rhombus. Then, find the value of x and y?
a)
40 and 40
b)
35 and 35
c)
37 and 37
d)
45 and 45
|
Parina Patel answered |
To find the values of x and y in the given rhombus ABCD, we need to understand the properties of a rhombus. A rhombus is a quadrilateral with all sides of equal length.
- The diagonals of a rhombus bisect each other at right angles. This means that the diagonals intersect at a 90-degree angle.
- The diagonals of a rhombus divide it into four congruent triangles.
- The opposite angles in a rhombus are equal.
Given these properties, let's solve for x and y.
First, let's consider the diagonals of the rhombus. Let the intersection point of the diagonals be O.
- The diagonals AC and BD bisect each other at right angles, so angle AOC and angle BOD are right angles.
Since the diagonals divide the rhombus into congruent triangles, we can focus on one of these triangles, such as triangle AOB.
- In triangle AOB, angle AOB is equal to 90 degrees (as it is a right angle).
- The opposite angles in a rhombus are equal, so angle AOB is also equal to angle ABO.
Since the sum of angles in a triangle is 180 degrees, we can write the following equation:
angle AOB + angle ABO + angle BAO = 180 degrees
We know that angle AOB = 90 degrees, and angle AOB = angle ABO, so we can rewrite the equation as:
90 degrees + angle AOB + angle BAO = 180 degrees
Simplifying further, we get:
2 * angle AOB + angle BAO = 180 degrees
Since angle AOB = angle ABO, we can substitute angle AOB with x and angle BAO with y:
2x + y = 180 degrees
Now, let's consider the fact that opposite angles in a rhombus are equal. Since angle AOC is a right angle, angle AOD is also a right angle.
- The sum of angles in a triangle is 180 degrees, so we can write:
angle AOD + angle ADO + angle OAD = 180 degrees
Since angle AOD = 90 degrees and angle ADO = angle OAD, we can rewrite the equation as:
90 degrees + angle ADO + angle OAD = 180 degrees
Simplifying further, we get:
2 * angle ADO + angle OAD = 180 degrees
Substituting angle ADO with x and angle OAD with y, we get:
2x + y = 180 degrees
This equation is the same as the one we obtained earlier. Therefore, we can conclude that x = y.
Now, let's consider the fact that the diagonals of a rhombus bisect each other at right angles. This means that the diagonals AC and BD are perpendicular.
Since opposite sides of a rhombus are parallel, the diagonals bisect each other into two congruent right-angled triangles. In each of these triangles, the sum of angles is 180 degrees. Therefore, we can write:
angle ADC + angle ACB + angle CDB = 180 degrees
Since angle ADC = angle ACB (as opposite angles are equal), we can rewrite the equation as:
2 * angle ADC + angle CDB = 180 degrees
Substituting angle ADC with x and
- The diagonals of a rhombus bisect each other at right angles. This means that the diagonals intersect at a 90-degree angle.
- The diagonals of a rhombus divide it into four congruent triangles.
- The opposite angles in a rhombus are equal.
Given these properties, let's solve for x and y.
First, let's consider the diagonals of the rhombus. Let the intersection point of the diagonals be O.
- The diagonals AC and BD bisect each other at right angles, so angle AOC and angle BOD are right angles.
Since the diagonals divide the rhombus into congruent triangles, we can focus on one of these triangles, such as triangle AOB.
- In triangle AOB, angle AOB is equal to 90 degrees (as it is a right angle).
- The opposite angles in a rhombus are equal, so angle AOB is also equal to angle ABO.
Since the sum of angles in a triangle is 180 degrees, we can write the following equation:
angle AOB + angle ABO + angle BAO = 180 degrees
We know that angle AOB = 90 degrees, and angle AOB = angle ABO, so we can rewrite the equation as:
90 degrees + angle AOB + angle BAO = 180 degrees
Simplifying further, we get:
2 * angle AOB + angle BAO = 180 degrees
Since angle AOB = angle ABO, we can substitute angle AOB with x and angle BAO with y:
2x + y = 180 degrees
Now, let's consider the fact that opposite angles in a rhombus are equal. Since angle AOC is a right angle, angle AOD is also a right angle.
- The sum of angles in a triangle is 180 degrees, so we can write:
angle AOD + angle ADO + angle OAD = 180 degrees
Since angle AOD = 90 degrees and angle ADO = angle OAD, we can rewrite the equation as:
90 degrees + angle ADO + angle OAD = 180 degrees
Simplifying further, we get:
2 * angle ADO + angle OAD = 180 degrees
Substituting angle ADO with x and angle OAD with y, we get:
2x + y = 180 degrees
This equation is the same as the one we obtained earlier. Therefore, we can conclude that x = y.
Now, let's consider the fact that the diagonals of a rhombus bisect each other at right angles. This means that the diagonals AC and BD are perpendicular.
Since opposite sides of a rhombus are parallel, the diagonals bisect each other into two congruent right-angled triangles. In each of these triangles, the sum of angles is 180 degrees. Therefore, we can write:
angle ADC + angle ACB + angle CDB = 180 degrees
Since angle ADC = angle ACB (as opposite angles are equal), we can rewrite the equation as:
2 * angle ADC + angle CDB = 180 degrees
Substituting angle ADC with x and
Consecutive angles of a Parallelogram are- a)Supplementary
- b)Acute
- c)Complementary
- d)Equal
Correct answer is option 'A'. Can you explain this answer?
Consecutive angles of a Parallelogram are
a)
Supplementary
b)
Acute
c)
Complementary
d)
Equal
|
|
Suhani Kumari answered |
Option A (supplementary)As one of the basic properties of parallelograms is that any pair of consecutive angles are supplementary.
Three statements are given below:
(I) In a Rectangle ABCD, the diagonals AC bisects ∠A as well as ∠C.
(II) In a Square ABCD, the diagonals AC bisects ∠A as well as ∠C.
(III) In rhombus ABCD, the diagonals AC bisects ∠Aas well as ∠C.
Which is True?- a)III
- b)I
- c)II and III
- d)II
Correct answer is option 'C'. Can you explain this answer?
Three statements are given below:
(I) In a Rectangle ABCD, the diagonals AC bisects ∠A as well as ∠C.
(II) In a Square ABCD, the diagonals AC bisects ∠A as well as ∠C.
(III) In rhombus ABCD, the diagonals AC bisects ∠Aas well as ∠C.
Which is True?
(I) In a Rectangle ABCD, the diagonals AC bisects ∠A as well as ∠C.
(II) In a Square ABCD, the diagonals AC bisects ∠A as well as ∠C.
(III) In rhombus ABCD, the diagonals AC bisects ∠Aas well as ∠C.
Which is True?
a)
III
b)
I
c)
II and III
d)
II
|
|
Prateek Kushwaha answered |
The diagonals of rectangle are not bisects angle....
But diagonals of square and rhombus are bisects..
so option C is correct
But diagonals of square and rhombus are bisects..
so option C is correct
A cyclic quadrilateral is such that two of its adjacent angles are divisible by 6 and 10 respectively. One of the remaining angles will necessarily be divisible by:
- a)3
- b)4
- c)8
- d)None of these
Correct answer is option 'D'. Can you explain this answer?
A cyclic quadrilateral is such that two of its adjacent angles are divisible by 6 and 10 respectively. One of the remaining angles will necessarily be divisible by:
a)
3
b)
4
c)
8
d)
None of these
|
|
Palak Bose answered |
We know that the sum of the opposite angles of a cyclic quadrilateral is 180 degrees. Let the four angles be A, B, C, and D, with A and B being the angles divisible by 6 and 10, respectively.
Since A is divisible by 6 and B is divisible by 10, we know that A = 6m and B = 10n for some integers m and n.
Now, consider the opposite angles. Since the sum of opposite angles is 180 degrees, we have:
C = 180 - B = 180 - 10n
D = 180 - A = 180 - 6m
We want to find which of the given options the angles C or D are necessarily divisible by. Let's examine each option:
1. 3: Since B is divisible by 10, it is possible that B is divisible by 5 but not 3 (e.g. B = 10). In this case, C = 180 - B would not be divisible by 3. Also, A is divisible by 6, so A is always divisible by 3, which means D = 180 - A would never be divisible by 3. So, this option is incorrect.
2. 4: Since A is divisible by 6, it is possible that A is divisible by 2 but not 4 (e.g. A = 6). In this case, D = 180 - A would not be divisible by 4. Also, B is divisible by 10, so B is always divisible by 2, which means C = 180 - B would never be divisible by 4. So, this option is also incorrect.
3. 8: If A is divisible by 6, then it can be even or odd multiples of 6 (e.g. A = 6, 12, 18, ...). D will be 180 - A, which means D can be both even and odd (e.g. D = 180 - 6 = 174, D = 180 - 12 = 168, D = 180 - 18 = 162, ...). Since D can be both even and odd, it is not necessarily divisible by 8. Similarly, C can also be both even and odd, so it is not necessarily divisible by 8. Thus, this option is also incorrect.
4. None of these: Since none of the previous options work, the correct answer is None of these.
So, the correct answer is option 4: None of these.
Since A is divisible by 6 and B is divisible by 10, we know that A = 6m and B = 10n for some integers m and n.
Now, consider the opposite angles. Since the sum of opposite angles is 180 degrees, we have:
C = 180 - B = 180 - 10n
D = 180 - A = 180 - 6m
We want to find which of the given options the angles C or D are necessarily divisible by. Let's examine each option:
1. 3: Since B is divisible by 10, it is possible that B is divisible by 5 but not 3 (e.g. B = 10). In this case, C = 180 - B would not be divisible by 3. Also, A is divisible by 6, so A is always divisible by 3, which means D = 180 - A would never be divisible by 3. So, this option is incorrect.
2. 4: Since A is divisible by 6, it is possible that A is divisible by 2 but not 4 (e.g. A = 6). In this case, D = 180 - A would not be divisible by 4. Also, B is divisible by 10, so B is always divisible by 2, which means C = 180 - B would never be divisible by 4. So, this option is also incorrect.
3. 8: If A is divisible by 6, then it can be even or odd multiples of 6 (e.g. A = 6, 12, 18, ...). D will be 180 - A, which means D can be both even and odd (e.g. D = 180 - 6 = 174, D = 180 - 12 = 168, D = 180 - 18 = 162, ...). Since D can be both even and odd, it is not necessarily divisible by 8. Similarly, C can also be both even and odd, so it is not necessarily divisible by 8. Thus, this option is also incorrect.
4. None of these: Since none of the previous options work, the correct answer is None of these.
So, the correct answer is option 4: None of these.
Read the passage below and solve the questions based on it The area of a square is equal lo the area of a rectangle. Moreover, the perimeter of the square is also equal to the perimeter of the rectangle,Q.The length of the rectangle is equal to the:a)Breadth of the rectangleb)Side of the squarec)Cannot be determinedd)Both [1] and [2]Correct answer is option 'D'. Can you explain this answer?
|
Nikita Singh answered |
This is possible only when both the length and breadth of the rectangle are equal to the side of the square.
Four horses are tethered at four comers of a square plot of side 14 m so that the adjacent horses can just reach one another. There is a small circular pond of area 20 m2 at the centre. Find the ungrazed area.- a)42 m2
- b)22 m2
- c)84 m2
- d)168 m2
Correct answer is option 'B'. Can you explain this answer?
Four horses are tethered at four comers of a square plot of side 14 m so that the adjacent horses can just reach one another. There is a small circular pond of area 20 m2 at the centre. Find the ungrazed area.
a)
42 m2
b)
22 m2
c)
84 m2
d)
168 m2
|
|
Preeti Khanna answered |
Total area of plot = 14 * 14 = 196m2
Horses can graze in quarter circle of radius = 7m
Grazed area = 4 * (pie r2)/4 = 154 m2
Area of plot when horses cannot reach = (196 - 154) = 42m2
Ungrazed area = 42 - 20 = 22m2
Horses can graze in quarter circle of radius = 7m
Grazed area = 4 * (pie r2)/4 = 154 m2
Area of plot when horses cannot reach = (196 - 154) = 42m2
Ungrazed area = 42 - 20 = 22m2
The ratio of the sides of Δ ABC is 1:2:4. What is the ratio of the altitudes drawn onto these sides?- a)4:2:1
- b)1:2:4
- c)1:4:16
- d)None of these
Correct answer is option 'D'. Can you explain this answer?
The ratio of the sides of Δ ABC is 1:2:4. What is the ratio of the altitudes drawn onto these sides?
a)
4:2:1
b)
1:2:4
c)
1:4:16
d)
None of these
|
|
Aakash Giery answered |
Sum of any two sides should be greater than third side.
here 1+2=3 is not less than 4 ,
1+2<4 ,so="" triangle="" is="" not="" possible.="" ,so="" triangle="" is="" not="">
here 1+2=3 is not less than 4 ,
1+2<4 ,so="" triangle="" is="" not="" possible.="" ,so="" triangle="" is="" not="">
Given a triangular prism, then what can we conclude about the lateral faces?
- a)Faces are rectangle
- b)Faces are Trapezium
- c)Faces are Prism
- d)both rectangles Faces are Parallelogram
Correct answer is option 'D'. Can you explain this answer?
Given a triangular prism, then what can we conclude about the lateral faces?
a)
Faces are rectangle
b)
Faces are Trapezium
c)
Faces are Prism
d)
both rectangles Faces are Parallelogram
|
|
Sarita Reddy answered |
In a triangular prism, the bases are triangles and the lateral faces are parallelograms.
Thus it can be both rectangles and parallelograms.
So, option (D) is correct.
If the faces are rectangles, then we call it as right triangular prism
Observe the following figure.
It is clear that the faces of the above figure are rectangles and hence it is right triangular prism.
In the given figure, AB and CD are two chords of a circle intersecting at O. If AO = 4 cm and OB = 6 cm and OC = 3 cm. Find OD.
- a)4 cm
- b)6 cm
- c)8 cm
- d)10 cm
Correct answer is option 'C'. Can you explain this answer?
In the given figure, AB and CD are two chords of a circle intersecting at O. If AO = 4 cm and OB = 6 cm and OC = 3 cm. Find OD.
a)
4 cm
b)
6 cm
c)
8 cm
d)
10 cm
|
Ishita Reddy answered |
By the rule of chords, cutting externally, we get:
(9 + 6) * 6 = (5 + x) * 5
90 = 25 + 5x
5x = 65
x = 13 cm
90 = 25 + 5x
5x = 65
x = 13 cm
.
D, E, F are the mid points of the sides BC, CA and AB respectively of ΔABC. Then ΔDEF is congruent to triangle- a)ABC
- b)AEF
- c)BFD, CDE
- d)AFE, BFD, CDE
Correct answer is option 'D'. Can you explain this answer?
D, E, F are the mid points of the sides BC, CA and AB respectively of ΔABC. Then ΔDEF is congruent to triangle
a)
ABC
b)
AEF
c)
BFD, CDE
d)
AFE, BFD, CDE
|
Eesha Bhat answered |
The correct option is Option D.
Given : D, E and F are tge mid-point of the sides BC, CA and AB respectively of Δ ABC.
To prove : Δ DEF is congruent to traingle
Proof :
Since E and F are midpoints of AC and AB.
BC II FE and FE = ½ BC = BD (By mid point theorem)
BD II FE and BD = FE
Similarly, BF II DE and BF = DE
Hence, BDEF is a parallelogram (A pair of opposite sides are equal and parallel)
Similarly, we can prove that FDCE and AFDE are also parallelograms.
Now, BDEF is a parallelogram so it's diagonal FD divides it into two traingles of equal areas.
Therefore, ar(Δ BDF) = ar(Δ DEF).......... (i)
In parallelogram AFDE,
ar(Δ AFE) = ar(Δ DEF) (EF is a diagonal)......... (ii)
In parallelogram FDCE,
ar(Δ CDE) = ar(Δ DEF) (DE is a diagonal)...........(iii)
From (i), (ii) and (iii)
ar(Δ BDF) = ar(Δ AFE) = ar(Δ CDE) = ar(Δ DEF)..........(iv)
If area of traingles are equal then they are congruent.
Hence, Δ DEF is congruent to triangle Δ BDF = Δ AFE = Δ CDE.
In an equilateral triangle, the incentre, circumcentre, orthocentre and centroid are:- a)concylic
- b)coincident
- c)collinear
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
In an equilateral triangle, the incentre, circumcentre, orthocentre and centroid are:
a)
concylic
b)
coincident
c)
collinear
d)
none of these
|
Gaurav Kumar answered |
The centroid is the intersection of the three medians while the incentre is the intersection of the three (internal) angle bisectors. In an equilateral triangle, each median is also an angle bisector (and vice versa), the centroid coincides with the incentre. In fact, the centroid, incentre, circumcentre and orthocentre of an equilateral triangle are coincide at the same point.
The area of similar triangles, ABC and DEF are 144cm2 and 81 cm2 respectively. If the longest side of the larger △ABC be 36 cm, then the longest side of the smaller △DEF is:
- a)27 cm
- b)26 cm
- c)29 cm
- d)30 cm
Correct answer is option 'A'. Can you explain this answer?
The area of similar triangles, ABC and DEF are 144cm2 and 81 cm2 respectively. If the longest side of the larger △ABC be 36 cm, then the longest side of the smaller △DEF is:
a)
27 cm
b)
26 cm
c)
29 cm
d)
30 cm
|
|
Aarav Sharma answered |
Given:
- The area of triangle ABC is 144 cm^2.
- The area of triangle DEF is 81 cm^2.
- The longest side of triangle ABC is 36 cm.
To find:
- The length of the longest side of triangle DEF.
Explanation:
Step 1: Finding the Scale Factor
- The area of a triangle is given by the formula: Area = (1/2) * base * height.
- Since the triangles ABC and DEF are similar, their areas are proportional to the square of their corresponding sides.
- Therefore, we can write: (AB/DE)^2 = Area of ABC/Area of DEF.
- Substituting the given values, we get: (AB/DE)^2 = 144/81.
- Simplifying the equation, we have: (AB/DE)^2 = 16/9.
- Taking the square root of both sides, we get: AB/DE = √(16/9).
- Simplifying further, we have: AB/DE = 4/3.
- This ratio represents the scale factor between the two triangles.
Step 2: Finding the Length of the Longest Side of Triangle DEF
- The longest side of triangle ABC is given as 36 cm.
- Using the scale factor, we can write: AB/DE = 4/3.
- Substituting the values, we have: 36/DE = 4/3.
- Cross-multiplying, we get: 4DE = 36 * 3.
- Simplifying the equation, we have: 4DE = 108.
- Dividing both sides by 4, we get: DE = 27.
- Therefore, the length of the longest side of triangle DEF is 27 cm.
Conclusion:
- The length of the longest side of triangle DEF is 27 cm.
- Hence, the correct answer is option A.
- The area of triangle ABC is 144 cm^2.
- The area of triangle DEF is 81 cm^2.
- The longest side of triangle ABC is 36 cm.
To find:
- The length of the longest side of triangle DEF.
Explanation:
Step 1: Finding the Scale Factor
- The area of a triangle is given by the formula: Area = (1/2) * base * height.
- Since the triangles ABC and DEF are similar, their areas are proportional to the square of their corresponding sides.
- Therefore, we can write: (AB/DE)^2 = Area of ABC/Area of DEF.
- Substituting the given values, we get: (AB/DE)^2 = 144/81.
- Simplifying the equation, we have: (AB/DE)^2 = 16/9.
- Taking the square root of both sides, we get: AB/DE = √(16/9).
- Simplifying further, we have: AB/DE = 4/3.
- This ratio represents the scale factor between the two triangles.
Step 2: Finding the Length of the Longest Side of Triangle DEF
- The longest side of triangle ABC is given as 36 cm.
- Using the scale factor, we can write: AB/DE = 4/3.
- Substituting the values, we have: 36/DE = 4/3.
- Cross-multiplying, we get: 4DE = 36 * 3.
- Simplifying the equation, we have: 4DE = 108.
- Dividing both sides by 4, we get: DE = 27.
- Therefore, the length of the longest side of triangle DEF is 27 cm.
Conclusion:
- The length of the longest side of triangle DEF is 27 cm.
- Hence, the correct answer is option A.
The length of each side of a rhombus is 10cm and one of its diagonal is of length 16cm. The Length of the other Diagonal is:- a)5cm
- b)12cm
- c)13cm
- d)6cm
Correct answer is option 'B'. Can you explain this answer?
The length of each side of a rhombus is 10cm and one of its diagonal is of length 16cm. The Length of the other Diagonal is:
a)
5cm
b)
12cm
c)
13cm
d)
6cm
|
|
Abhishek Kumar answered |
Visualise a rhombus of each side 10 cm.
Now, we know that the diagonals of a rhombus are perpendicular bisectors i.e it is perpendicular to each other and divide each diagonal into two parts
since there are two diagonal so
there will be four parts out of which two parts of the same diagonal are equal .
So the diagonal of length 16cm is divided into two parts each of 8 cm
Since two diagonal are perpendicular bisectors of each other so it will divide the rhombus into four equal right angled triangle
visualising one triangle we get:
Hypotenuse =side of rhombus =10cm
one side whose measure os unknown =x (say)
another side =8 cm.
Applying phythogorus theorem
hypotenuse ² = base² (one side)+ perpendicular²
or,
hypotenuse²-base² =perpendicular ²
10² - 8²= perpendicular²
100-64 = perpendicular ²
36=perpendicular ²
6²=perpendicular ²
6=perpendicular
since diagonal is divided into two parts
total length of one diagonal =2×perpendular.
=2×6=12cm
The Diagonals AC and BD of a Parallelogram ABCD intersect each other at the point O such that ∠DAC = 30∘ and ∠AOB = 70∘. Then, ∠DBC?- a)30∘
- b)45∘
- c)35∘
- d)40∘
Correct answer is option 'D'. Can you explain this answer?
The Diagonals AC and BD of a Parallelogram ABCD intersect each other at the point O such that ∠DAC = 30∘ and ∠AOB = 70∘. Then, ∠DBC?
a)
30∘
b)
45∘
c)
35∘
d)
40∘
|
|
Akanksha Choudhury answered |
AO = 5, OC = 7, and OD = 9. Find the length of OB.
We know that in a parallelogram, opposite sides are equal in length, so AB = CD and BC = AD. Let x be the length of OB.
By the properties of diagonals in parallelograms, we know that AO/OC = BO/OD. Substituting the given values, we have:
5/7 = x/9
Solving for x, we get:
x = 9(5/7) = 6.43 (rounded to two decimal places)
Therefore, the length of OB is approximately 6.43 units.
We know that in a parallelogram, opposite sides are equal in length, so AB = CD and BC = AD. Let x be the length of OB.
By the properties of diagonals in parallelograms, we know that AO/OC = BO/OD. Substituting the given values, we have:
5/7 = x/9
Solving for x, we get:
x = 9(5/7) = 6.43 (rounded to two decimal places)
Therefore, the length of OB is approximately 6.43 units.
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Mathematics & Pedagogy Paper 2 for CTET & TET Exams
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