Test: Triangles- 2

Given: Some number x

• ΔLMN with sides as shown below:

Find: Is ΔLMN an acute triangle?

Step 3: Analyze Statement 1 independently

• NM = 10
• That is, |x -4| = 10
  • So, x is a number whose distance from 4 on the number line is 10

• So, x = -6 or 14
  • Rejecting the negative value since x + 1, being the length of a geometric figure, must be positive.
• So, x = 14
• Therefore, the sides of the triangle are:
  • 15, 10, 7
    • The longest side is 15
  • 10² + 7² < 15²
• Therefore, the triangle is obtuse.
  • (Note: An obtuse triangle is a triangle one of whose angles is greater than 90^∘)
• So, the answer to the given question is NO.
• Since we’ve been able to arrive at a unique answer to the question, Statement 1 alone is sufficient.

Step 4: Analyze Statement 2 independently

• LM = 7
• |x-7| = 7
  • So, x is a number whose distance from 7 on the number line is 7

• So, x = 0 or 14
  • Case 1: x = 0
  • The sides of the triangle are:
    • 1, 4, 7
      • The longest side is 7
  • In this case, the sum of 2 sides of the triangle (1 + 4) is not greater than the 3^rd side
  • Therefore, this triangle is not possible
  • So, this value of x is not possible.

• Case 2: x = 14
• So, the sides of the triangle are:
  • 15, 10, 7
    • The longest side is 15
  • 10² + 7² < 15²

• Therefore, the triangle is obtuse.
• So, the answer to the given question is NO.
• Since we’ve been able to arrive at a unique answer to the question, Statement 2 alone is sufficient

• 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 .

Steps 1 & 2: Understand Question and Draw Inferences

Given: ?ABC

To find: Is AC² = AB² + BC²?

• The answer will be YES if ∠ABC=90∘,

Step 3: Analyze Statement 1 independently

Statement 1 says that ‘∠BAC + ∠ACB = ∠ABC’

• Using Angle Sum Property in ?ABC, we can write: ∠BAC+?ACB+?ABC=180∘

• Substituting the Statement 1 equation above:  ∠ABC+∠ABC=180∘
• 2(∠ABC)=180^∘
• ∠ABC=90^∘
• Therefore, the answer to the question is YES
• Since we’ve been able to find a unique answer to the question, Statement 1 alone is sufficient

Step 4: Analyze Statement 2 independently

Statement 2 says that AB = BC

• This means that ?ABC is isosceles
• However, not every isosceles triangle is right-angled
• Therefore, knowing only that AB = BC is clearly not sufficient to confirm that ?ABC is right-angled at B
• So, Statement 2 is not sufficient.

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

Given:

• Triangles AED and BEC are formed using the straight lines AB and CD
  • So, ∠AED = ∠CEB = x (Vertically opposite angles)

• In ΔBEC,
  • BE = BC
  • So, ∠CEB = ∠ECB = x
  • Therefore, ∠EBC = 180^∘−2x

• In ΔAED,
  • ∠EAD = 180^∘−(x+2x)=180^∘−3x

• We are given that DE² > AD² + AE²
  • This means, ΔAED is obtuse
    • So, ∠EAD > 90^∘
    • 180^∘−3x>90^∘
    • 90^∘>3x
    • That is, x<30∘

To find: Which of the 3 statements must be true?

Approach:

1. Based on the given and inferred information above, we’ll evaluate the 3 statements one by one

Working Out:

• Evaluating Statement I
  • Statement I says that CE² > 2BE²
    • That is, CE² > BE² + BC² (since we are given that BE = BC)
  • This inequality will be true if ∠EBC is obtuse.
  • Checking if ∠EBC is obtuse
    • ∠EBC = 180^∘−2x
    • Inferred above: x<30^∘
    • So,−2x>−60^∘  (multiplying both sides with a negative number reverses the sign of inequality)
    • 180^∘–2x>180^∘–60^∘
    • 180^∘–2x>120^∘ 
    • That is, ∠EBC>120^∘

• Since ∠EBC is indeed obtuse, Statement I will be true for all values of x

• Evaluating Statement II
  • Statement II says that AE < AD
  • Checking if this statement is true:
    • In ΔAED, the angles in increasing order of magnitude are:
      • ∠AED < ∠ADE < ∠DAE
        • Note that since we’ve already inferred that ∠DAE is an obtuse angle, it will definitely be the greatest angle of this triangle. Among the remaining 2 angles, the angle that measures 2x will obviously be greater than the angle that measures x.

• So, the sides of this triangle in increasing order of magnitude will be:
  • AD < AE < DE
• Thus, Statement II is NOT true.

• Evaluating Statement III
  • Statement III says that DE < CE
  • This inequality compares the sides of two different triangles. We do not have enough information to make this comparison. So, it is not a must be true statement based on the limited information that we have.

• Getting to the final answer
  • Thus, only Statement I is a must be true statement.

Looking at the answer choices, we see that the correct answer is Option A

Given:

• An obtuse triangle, 2 of whose sides are AB = 3 and BC = 4 units

To find:

• The possible measure of the side AC that is opposite to the obtuse angle

Approach & Working:

• To answer the question, we need to think of the constraints on AC. From the theory of triangles, we can figure that there are 2 constraints on AC:
  • Constraint 1: Since triangle ABC is obtuse-angled at B, AC² > AB² + BC²
  • Constraint 2: The sum of any 2 sides of a triangle must be greater than the 3^rd side. So, AB + BC > AC
• So, we will work out the range of possible values of AC by applying these 2 constraints, and then see, which of the 3 given values fit this range.
• Applying Constraint 1
  • AC² > 3² + 4²
  • AC² > 9 + 16
  • AC² > 25
  • So, AC > 5
• Applying Constraint 2
  • AB + BC > AC
  • So, 3 + 4 > AC
  • 7 > AC
  • Rewritten as: AC < 7
• Getting to the answer
  • Thus, we see that 5 < AC < 7
  • The only value, out of the 3 options, that satisfies this range is Option I

Thus, the correct answer is Option A

Given:

• Length of rod = 10 meters
• Angle with the ground at 1^st instant = 60^o
• Angle with the ground at 2^nd instant = 30^o

To find:

• Vertical distance covered between the 2 instants

Approach:

• We will first try to understand the question
• To do so, let us:
  • Drop perpendiculars from the free end of the rod at both instants of time
  • Label the triangles thus formed

• From the diagram, it is clear that we need to find AB – PQ
  • (Note: if you are not sure how AB – PQ is equal to the vertical distance covered by the ladder, drop a perpendicular PX on AB. You’ll see that BQPX is a rectangle and so, PQ = BX. So, AB – PQ = AB – BX = AX. The ladder was earlier at a height of A and it is now at a height of X. So, the distance AX clearly denotes the vertical distance covered by the ladder.)

1. Thus, we need to find AB – PQ
   1. AB is a side of right ΔOBA, which is 30-60- 90 Triangle and angle AOB = 60^o
   2. PQ is a side of right ΔOQP, which is also a 30-60-90 Triangle with angle POQ = 30^o
2. In each of the 2 right triangles, we know a side and two angles.
   1. So, by using side ratio property of 30-60-90 triangle, we can find the required values

Working Out:

• Finding AB
  • In right ΔOBA
    • OB: AB: OA = 1: √3: 2 
    • We know OA = 10
    • AB: OA = √3: 2
    • Thus AB = 5√3

• Finding PQ
  • In right ΔOQP
    • PQ: OQ: PO = 1: √3: 2 
    • We know PO = 10
    • PQ: PO = 1: 2
    • Thus PQ = 5
• Thus, AB – PQ = 5√3 – 5 = 5 (√3 -1) meters
• Therefore, the correct answer is Option A.

Steps 1 & 2: Understand Question and Draw Inferences

Given:

• Isosceles ΔABC
  • AB = AC
  • So, perpendicular AD will bisect side BC.
  • If we assume ∠ ABC to be x^o, the different ∠s in the figure can be represented as below:

To find: Perimeter of ΔABC

• =AB + BC + CA
• = 2AB + BC

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.

• From Statement 2 

• Using Equation II, a unique value of AD will be obtained
• Using Equation I, the value of 2AB + BC will be obtained

Sufficient.

Answer: Option C

Steps 1 & 2: Understand Question and Draw Inferences

Given:

To find:

• Area of quadrilateral PQRS
  • Area of quadrilateral PQRS= Area of Right ΔPSR + Area of Right ΔPRQ

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

• x = 3 or 4  (Rejected the negative roots because, being the length of a triangle, x cannot be negative)
• Since we didn’t get a unique value of x,
  • Statement 1 is not sufficient to answer the question.

Step 4: Analyze Statement 2 independently

(2) The area of ΔPRQ is 30 square units

• Since the value of x is unique it is sufficient to answer the question.
• Hence Statement 2 is sufficient to find the answer.

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.

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

• Let’s drop a perpendicular CD on side AB.

• But the question is, is D the mid-point of AB?

We do not know.

Therefore, Statement 1 is not sufficient to answer the question.

Step 4: Analyze Statement 2 independently

• Point C is the midpoint of EF
• Let’s drop a perpendicular CD on side AB

• CD is parallel to and equal to BE.
  • Since C is the mid-point of EF, D will be the mid-point of AB.

• Therefore, AD = AB/2

• But, we don’t know the magnitude of either AB or CD or AC. So, we cannot find the angles of the triangle.

Not sufficient.

Step 5: Analyze Both Statements Together (if needed)

• From Statement 1: If D is the mid-point of AB, then triangle ABC is an equilateral triangle
• From Statement 2: D is the mid-point of AB

Thus, the two statements together are sufficient to answer the question.

Answer: Option C

Given:

• Isosceles ΔABC
  • AB = BC
  • BD perpendicular to AC
    • Since this is an isosceles Δ, BD will also bisect AC
      • So, AD = CD

• Let’s depict the given and deduced information in the diagram:

• Equal sides AB and BC are depicted in orange
• Equal sides AD, ED and DC are depicted in green

We have considered: CD = x

• Since CD = ED = AD
  • We can denote each of them with “x”
• Also, BD = BE + ED
  • BE = BD – ED
  • BE = √3x – x = (√3 – 1) x
• Also AC = AD + CD = x + x = 2x
• Substituting the value of AC, BE, BD and CD in equation (i) and (ii) we get
  • Area of ΔBEC = ½ * BE * CD = ½ * (√3 – 1) x * x
  • Area of ΔABC = ½ * 2x * √3x
• Thus the ratio of (Area of ΔBEC)/(Area of ΔABC) is equal to

Therefore, the correct answer is Option A.

Given:

• Right ΔABC
  • ∠ACB = 45^∘
  • AC = 3√2

• Right ΔABC
  • ∠CBD = 30^∘
  •  

To find: area of ΔBCD

Working Out:

• Finding BC

Finding CD

Given

• In ΔABD, the sides of the triangle are: {2, 2, x}
• In ΔBDC, the sides of the triangle are: {x, 2, 4}
• x is an integer
  • Clearly, since x denotes the length of a side of a triangle, x must be a positive integer.

To Find : x= ?

Approch :

• To answer the question, we need to find the constraints on x.
• Since we’re given the other side lengths of the 2 triangles, we’ll apply the property that the sum of 2 sides of a triangle is always greater than its third side, on both ΔABD and ΔBDC to get to a constraint on x.

Working out

• Applying the property in ΔABD
  • 2 + 2 > x
    • That is, x < 4
  • And, 2 + x > 2
    • That is, x > 0
  • Thus, possible values of x: {1, 2, 3}

• Applying the property in ΔBDC
  • 4 + 2 > x
    • That is, x < 6
  • And, 2 + x > 4
    • That is, x > 2
  • Thus, possible values of x: {3, 4, 5}
• Getting to the answer
  • By considering ΔABD, we got: x = {1, 2, 3}
  • By considering ΔBDC, we got: x = {3, 4, 5}
  • The only value of x that satisfies the property (that 2 sides of a triangle are greater than the 3 side) for both triangles is x = 3
  • Therefore, the value of x is 3

Looking at the answer choices, we see that the correct answer is Option B 

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

• OA = x + 1
  • In triangle AOB, x + x+1 = 5 (By Pythagoras Theorem)

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

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 = ?

• As per the above figure, ∠ACB = 180 — (x° + y°)
• So, in order to find its value, we need to know the value of X ° + y °

Step 3: Analyze Statement 1 independently

• ∠CAD = 90°
• That is, X° + y° = 90 . Sufficient

Step 4: Analyze Statement 2 independently

• ∠ CBD = 60°
• + y ° =   60°

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

Given

• A ladder inclined at an angle of 60 to the ground as shown below:

Let the height from which the coin falls be h meter
Time taken to fall =   seconds

To Find : Time taken to fall

Approach

• We need to find the time taken by the coin to fall
  • Time taken to fall = seconds
  • Since the value of time taken depends on h, we will need to find the value of h first
    • The ladder, ground and the wall form a right triangle
    • Since we know a side and an angle in this triangle, we can find h by using trigonometric ratios

Woeking out:

Since the right angled triangle is a 30-60-90 Triangle, we can write -

Steps 1 & 2: Understand Question and Draw Inferences

To find: the value of a

Step 3: Analyze Statement 1 independently

• Area of an equilateral Δ of side length 3a is 9√3
•  

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

• The area of an isosceles triangle whose sides are of length 4a, 4a and a units is 3√7 square units
• Let’s draw the given triangle:

• A perpendicular on the side that measures a units from the opposite vertex will bisect this side (Property of Isosceles triangles)
  • Let the height of the perpendicular be h
• So, by applying Pythagoras Theorem,

=>   (Rejecting the negative value of h since the length of a perpendicular cannot be negative)

so a² =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
 

step 1 & 2: Understand Question and Draw Inference
Given:

Representing the given information visually as below:

• In triangle BCD, CD = CB
  • So, angles opposite to these sides will be equal. Let these angles measure X^o
• Let ∠ACD = y^∘
• So, the different angles in this figure will be as under:
•  

To find: If AD > BC

• That is, if AD > CD (since BC = CD)
• That is, if in triangle ADC, 
  • ∠ACD > ∠DAC (since sides opposite to greater angles are greater)
  • That is, y^∘ > x^∘ − y^∘
  • That is, 2y^∘ > x^∘
• So, the answer to the question will be YES if
• 2y^∘ > x^∘

​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

Since, Δ ABC and Δ PQR are similar triangles.
then,∠ B = ∠ Q = 83⁰
Thus, in Δ ABC,
∠ C = 180⁰ - (∠ A + ∠ B)
or, ∠ C = 180⁰ - (47⁰ + 83⁰)
∠ C = 50⁰.

In Triangle ABD,
∠BAD + ∠B +90⁰ = 180⁰
or, ∠BAD + ∠ B = 90⁰ ------- (1)

Now, in Triangle ABC,
∠ACB + ∠B + ∠A = 180⁰
∠ACB + ∠B = 180⁰ - 90⁰ 
∠ACB = 90⁰ - ∠B -----(2)
From (1) and (2) , ∠BAD = ∠ACB 
So, AC = CD.

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.

Since, AB and CD bisects each other at O,
Hence, BC = AD = 6 cm.