In ΔLMN, LM = |x-7|, MN = |x-4| and NL = x + 1, where x is a number whose value is not known. Is ΔLMN an acute triangle (a triangle each of whose angles measures less than 90^{∘})?
(1) NM = 10
(2) LM = 7
Given: Some number x
Find: Is ΔLMN an acute triangle?
Step 3: Analyze Statement 1 independently
Step 4: Analyze Statement 2 independently
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in each of Steps 3 and 4, this step is not required
Hence the correct answer is Option D .
In ?ABC, is AC^{2} = AB^{2} + BC^{2}?
Steps 1 & 2: Understand Question and Draw Inferences
Given: ?ABC
To find: Is AC^{2} = AB^{2} + BC^{2}?
Step 3: Analyze Statement 1 independently
Statement 1 says that ‘∠BAC + ∠ACB = ∠ABC’
Step 4: Analyze Statement 2 independently
Statement 2 says that AB = BC
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 3, this step is not required
Answer: Option A
Triangles AED and BEC are formed using the straight lines AB and CD as shown in the figure above. If BE = BC, DE^{2} > AD^{2} + AE^{2} and the measure of ∠AED is x, which of the following statements must be true?
Given:
To find: Which of the 3 statements must be true?
Approach:
Working Out:
Looking at the answer choices, we see that the correct answer is Option A
The figure above shows a triangle ABC in which the measure of ∠ABC, b∘is greater than 90∘
If AB = 3 units and BC = 4 units, which of the following can be the measure of side AC?
Given:
To find:
Approach & Working:
Thus, the correct answer is Option A
The figure above shows the position, at two instants of time, of a 10-meter-long rod as it was falling down to the ground. If one end of the rod was pinned to the ground, by what vertical distance did the rod fall between the two instants of time?
Given:
To find:
Approach:
Working Out:
Triangle ABC is an isosceles triangle with AB = AC and AD is the perpendicular dropped from vertex A on the side BC. What is the perimeter of triangle ABC?
(1) ∠BAD = 2∠ACD
(2) The perimeter of triangle ADB is 15 + 5√3
Steps 1 & 2: Understand Question and Draw Inferences
Given:
To find: Perimeter of ΔABC
So, in order to find the perimeter, we need to know AC and BC
Knowing the ∠s may help us (because we may then employ trigonometric ratios)
Step 3: Analyze Statement 1 independently
(1) ∠BAD = 2∠ACD
90°- x° = 2x°
⇒ 3x° = 90°
⇒ x° = 30°
But we don’t know the magnitude of any side of the triangle. So, we cannot yet use trigonometric ratios to find the sides of the triangle.
Not sufficient.
Step 4: Analyze Statement 2 independently
(2) The perimeter of triangle ADB is 15 + 5√3
The value of 2AB + BC depends on the value of AD.
Since we do not know the value of AD, we cannot find the required value
Not sufficient.
Sufficient.
Answer: Option C
The figure above shows two right triangles ΔPSR and ΔPRQ that share a common side PR, which measures 5 units. If QR = 4PS, what is the area of the quadrilateral PQRS?
(1) The area of ΔPSR is 6 square units
(2) The area of ΔPRQ is 30 square units
Steps 1 & 2: Understand Question and Draw Inferences
Given:
To find:
So, in order to find the area, we need the value of x.
Step 3: Analyze Statement 1 independently
(1) The area of ΔPSR is 6 square units
Step 4: Analyze Statement 2 independently
(2) The area of ΔPRQ is 30 square units
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 4, this step is not required
And the correct answer is Option B.
Triangle ABC is inscribed in a rectangle ABEF forming two right triangles: AFC and BEC. Is triangle ABC an equilateral triangle?
Steps 1 & 2: Understand Question and Draw Inferences
Given:
The given information corresponds to the following figure:
To find: Is triangle ABC equilateral?
Step 3: Analyze Statement 1 independently
We do not know.
Therefore, Statement 1 is not sufficient to answer the question.
Step 4: Analyze Statement 2 independently
Not sufficient.
Step 5: Analyze Both Statements Together (if needed)
Thus, the two statements together are sufficient to answer the question.
Answer: Option C
In the figure above, triangle ABC is an isosceles triangle with AB = BC and BD is the perpendicular dropped from vertex B to the side AC. Point E is marked on BD such that ∠DAE = 45^{0} and ∠ECB = 15^{0}. If the ratio of length CD: BD = 1: √3. What is the ratio of the area of triangle BEC to the area of the triangle ABC?
Given:
We have considered: CD = x
Therefore, the correct answer is Option A.
In the figure above, ΔABC and ΔBCD are right-angled at points B and C respectively. If AC = 3√2, ∠ACB = 45^{∘} and ∠CBD = 30^{∘}, what is the area of ΔBCD?
Given:
To find: area of ΔBCD
Working Out:
Finding CD
In the figure shown above, x is the length of side BD of triangle ABD. If D is a point that lies on line AC and x is an integer, what is the value of x?
Given
To Find : x= ?
Approch :
Working out
Looking at the answer choices, we see that the correct answer is Option B
In the figure above, Δ POQ and Δ AOB are both right-angled at point O. If BQ = OB and OA = AP, and the length of side AB is 5 units, what is the area of the shaded region?
(1) OA = OB + 1
(2) If a rectangle is drawn with sides of length 2QB and PA /4 the area of the rectangle will be 6 square units
Given:
Right Δ POQ, in which:
BQ = OB
Means, B is the mid-point of OQ
OA = AP
Means, A is the mid-point of OP
AB = 5
To find:
Area of shaded region
= Area of Triangle POQ – Area of Triangle AOB
So, in order to answer the question, we need to know the value of OA and OB, or at least of the product OA*OB.
Step 3 : Analyze Statement 1 independently
Rejecting the negative value since the length of a geometric figure cannot be negative
Thus Statement 1 alone is sufficient to answer the question
Step 4 : Analyze Statement 2 independently
If a rectangle is drawn with sides of length 2QB and PA/4 the area of the rectangle will be 6 square units
Thus Statement 2 alone is also Sufficient to answer the question.
Step 5 : Analyze Both Statements together (if needed)
Since we’ve already arrived at a unique answer in each of Steps 3 and 4, this
step is not required
Answer: Option D
In the above figure, lines AB and CD are perpendicular to one another. If AD is parallel to BC and ∠CAB = 2∠ABD, what is the
measure of ∠ACB?
(1) ∠CAD = 90
(2) ∠ CBD = 60
Steps 1 & 2: Understand Question and Draw Inferences
Given:
• Let ∠ BAC = x°
• And ∠BAD = y°
• So, as per the given information, the different angles in this figure will look as below:
• In right AAEC, ∠ACE = 90 — x°(Angle Sum Property)
• In right AAED, ∠ ADE =90 — y (Angle Sum Property)
• Since AD is parallel to BC,
o ∠BCE = ∠ADE =90 — y° (Interior Alternate Angles)
o ∠CBA = ∠BAD = y°(Interior Alternate Angles)
• Given: ∠CAB = 2∠ABD
That is, x° = 2∠ABD
o Therefore, ∠ ABD =
O In right ABED, ∠EDB = 90° -
To find: ∠ACB = ?
Step 3: Analyze Statement 1 independently
Step 4: Analyze Statement 2 independently
Not sufficient to give us a unique value for X ° + y °.
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 3, this step is not required
Answer: Option A
Sam climbs up a 40/√3 metres long ladder that is inclined at an angle of 60 to the ground. Upon reaching the upper edge of the ladder, he drops a coin to the ground. If the time taken, in seconds, by an object to drop by a height of h meters is equal to , then how much time does the coin dropped by Sam take to reach the ground?
Given
Let the height from which the coin falls be h meter
Time taken to fall = seconds
To Find : Time taken to fall
Approach
Woeking out:
Since the right angled triangle is a 30-60-90 Triangle, we can write -
What is the value of a?
(1) The area of an equilateral triangle of side length 3a units is 9√3 square units
(2) The area of an isosceles triangle whose sides are of length 4a, 4a and a units is 3√7 square units
Steps 1 & 2: Understand Question and Draw Inferences
To find: the value of a
Step 3: Analyze Statement 1 independently
So, a = 2 (rejecting a = -2 because the side of a triangle, 3a, cannot be negative)
Thus, Statement 1 alone is sufficient to answer the question
Step 4: Analyze Statement 2 independently
=> (Rejecting the negative value of h since the length of a perpendicular cannot be negative)
so a^{2 }=4
That is, a = 2 (rejecting a = -2 because the side of a triangle, 3a, cannot be
negative)
Thus, Statement 2 alone is sufficient to answer the question
Step 5: Analyze both Statements together (if needed)
Since we’ve already arrived at a unique answer in each of Steps 3 and 4, this
step is not required
Answer: Option D
In ΔABC, point D lies on the side AB between vertices A and B.
If CD = CB, is AD > BC?
(1) ∠ADC = 110
(2) ∠ACD = 30
step 1 & 2: Understand Question and Draw Inference
Given:
Representing the given information visually as below:
To find: If AD > BC
Step 3 : Analyze Statement 1 independent
Step 4 : Analyze Statement 2 independent
Step 5: Analyze Both Statements Together (if needed)
From Statement 1:x = 70^{∘}
From Statement 2: y = 30^{∘}
Since we now know the values of x^{o }and y^{o }we can determine if 2y^{o }> X^{o}
Thus, the 2 statements together are sufficient to answer the question.
Answer: Option C
If ABC and PQR are similar triangles in which ∠ A = 47^{0} and ∠ Q = 83^{0}, then ∠ C is:
Since, Δ ABC and Δ PQR are similar triangles.
then,∠ B = ∠ Q = 83^{0}
Thus, in Δ ABC,
∠ C = 180^{0} - (∠ A + ∠ B)
or, ∠ C = 180^{0} - (47^{0} + 83^{0})
∠ C = 50^{0}.
In the following figure which of the following statements is true?
In Triangle ABD,
∠BAD + ∠B +90^{0} = 180^{0}
or, ∠BAD + ∠ B = 90^{0} ------- (1)
Now, in Triangle ABC,
∠ACB + ∠B + ∠A = 180^{0}
∠ACB + ∠B = 180^{0} - 90^{0}
∠ACB = 90^{0} - ∠B -----(2)
From (1) and (2) , ∠BAD = ∠ACB
So, AC = CD.
In triangle PQR length of the side QR is less than twice the length of the side PQ by 2 cm. Length of the side PR exceeds the length of the side PQ by 10 cm. The perimeter is 40 cm. The length of the smallest side of the triangle PQR is :
In Δ PQR,
QR +2 = 2PQ
QR = 2PQ - 2 ------- (2)
PR = PQ + 10 ----- (2)
Perimeter = 40 m
PQ + QR + Rp = 40
Putting the value of PQ and QR from equation (1) and (2),
Pq + 2PQ - 2 + PQ +10 = 40
4PQ = 32
PQ = 8 cm which is the smallest side of the triangle.
AB and CD bisect each other at O. If AD = 6 cm. Then BC is :
Since, AB and CD bisects each other at O,
Hence, BC = AD = 6 cm.
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