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Geometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT

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Question 1: x, y, z are integer that are side of an obtuse-angled triangle. If xy = 4, find z. 
A. 2
B. 3
C. 1
D. More than one possible value of z exists
Answer. (b)
Explanation. 
xy = 4
xy could be 2 * 2 or 4 * 1
221
222 These are the possible triangles.
223
441
22x will be a triangle if x is 1, 2 or 3 (trial and error).
44x is a triangle only if x is 1.
1. 221 is acute. 12 + 22 > 22
2. 222 is equilateral. So acute.
3. 223 is obtuse. 22 + 22 < 32
4. 144 is acute. 12 + 42 > 42
The question is " x, y, z are integer that are side of an obtuse-angled triangle. If xy = 4, find z."
And so, the third side has to be 3
Hence, the answer is 3.
Choice B is the correct answer.

Question 2How many isosceles triangles with integer sides are possible such that sum of two of the side is 12? 
A. 11
B. 6
C. 17
D. 23
Answer. (c)

Explanation.
Two possibilities: Two equal sides could add up to 12 or sum of 2 unequal sides = 12.
i.e. Sum of 2 equal sides = 12
Sum of 2 unequal sides = 12
⇒ If sum of two equal sides were 12, sides of the triangle should be 6, 6, x.
What are the values x can take?
x could range from 1 to 11.
11 integer values exist.
Now 2 unequal sides adding to 12. This could be 1 + 11, 2 + 10, 3 + 9, 4 + 8 or 5 + 7.
How many isosceles triangles are possible with the above combinations?
Isosceles triangles with the above combination:

1, 11, 11    ☑
1, 1, 11     ☒
2, 10, 10    ☑
2, 2, 10     ☒
3, 9, 9        ☑
3, 3, 9        ☒
4, 8, 8        ☑
4, 4, 8        ☒
5, 7, 7        ☑  
5, 5, 7        ☑

6 possibilities. Triplets such as (1, 1, 11), (2, 2, 10), etc are eliminated as sum of the two smaller values is less than the largest value. These cannot form a triangle.
The question is " How many isosceles triangles with integer sides are possible such that sum of two of the side is 12?"
11 + 6 = 17 possibilities totally.
Hence, the answer is 17.
Choice C is the correct answer.

Question 3: Sides of a triangle are 6, 10 and x for what value of x is the area of the △ the maximum? 

A. 8 cms

B. 9 cms

C. 12 cms

D. None of these
Answer. (d)
Explanation.
Side of a triangle are. 6, 10, x.
Area = 1/2 * 6 * 10 * sin∠BAC.
Area is maximum, when ∠BAC = 90◦
x = √(100+36) = √136
There is a more algebraic method using hero’s formula. Try that also.
The question is "Sides of a triangle are 6, 10 and x for what value of x is the area of the △ the maximum?"
Hence, the answer is None of these.
Choice D is the correct answer.

Question 4Two circles are placed in an equilateral triangle as shown in the figure. What is the ratio of the area of the smaller circle to that of the equilateral triangle 

Geometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT
A. π:36√3
B. π:18√3
C. π:27√3
D. π:42√3
Answer. (d)
Explanation.

Geometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT

In-radius of equilateral triangle of side a Geometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT
Diameter of larger circle Geometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT
Let us say common tangent PQ touches the two circle at R, center of smaller circle is I.
Now, PQ is parallel to BC. AR is perpendicular to PQ. Triangle PQR is also an equilateral triangle and AORID is a straight line. (Try to establish each of these observations. Just to maintain the rigour.) 

Geometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT
Radius of smaller circle = 1/3 radius of larger circle
Radius of smaller circle Geometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT
Area of smaller circle = πr2  
Geometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT
Geometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT  
Geometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT
The question is " What is the ratio of the area of the smaller circle to that of the equilateral triangle?"
Hence, the answer is π:27√3.
Choice C is the correct answer.

Question 5: Perimeter of a △ with integer sides is equal to 15. How many such triangles are possible? 
A. 7
B. 6
C. 8
D. 5
Answer. (a)
Explanation.
Let us assume a ≤ b ≤ c.
a = 1, Possible triangle 1, 7, 7
a = 2, possible triangle 2, 6, 7
a = 3, possible triangles 3, 6, 6 and 3, 5, 7
a = 4, possible triangles 4, 4, 7 and 4, 5, 6
a = 5, possible triangle is 5, 5, 5
There are totally 7 triangles possible.
The question is "Perimeter of a △ with integer sides is equal to 15. How many such triangles are possible?"
Hence, the answer is 7.
Choice A is the correct answer.

Question 6There is an equilateral triangle with a square inscribed inside it. One of the sides of the square lies on a side of the equilateral △. What is the ratio of the area of the square to that of the equilateral triangle? 
A. 12 : 12 + 7√3
B. 24 : 24 + 7√3
C. 18 : 12 + 15√3
D. 6 : 6 + 5√3
Answer. (a)
Explanation.

Geometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT

APQ is an equilateral △. As PQ is parallel to BC.
Let side of the square be ‘a’
AP = a = AQ
△QRC has angles 30 – 60 – 90. 
Geometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT
Geometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT
Geometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT
Area of equilateral Geometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT
Geometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT
Geometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT
Ratio of area of square to that of equilateral △ isGeometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT
= 12 : 12 + 7√3 
The question is "What is the ratio of the area of the square to that of the equilateral triangle?"
Hence, the answer is 12 : 12 + 7√3.
Choice A is the correct answer.

Question 7: △ABC has integer sides x, y, z such that xz = 12. How many such triangles are possible? 
A. 8
B. 6
C. 9
D. 12
Answer. (c)
Explanation.
xz = 12
x,z can be 1, 12 or 2, 6 or 3, 4
Possible triangles
1 - 12 - 12
2 - 6 - 5; 2 - 6 - 6; 2 - 6 - 7
3 - 4 - 2; 3 - 4 - 3; 3 - 4 - 4; 3 - 4 - 5; 3 - 4 - 6.
The question is "△ABC has integer sides x, y, z such that xz = 12. How many such triangles are possible?"
Hence, the answer is 9 triangles.
Choice C is the correct answer.

Question 8: ABCDE is a regular pentagon. O is a point inside the pentagon such that AOB is an equilateral triangle. What is ∠OEA? 

Geometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT
A. 66°
B. 48°
C. 54°
D. 72°
Answer. (a)
Explanation.

Geometry Questions with Answer- 1 Notes | Study Quantitative Aptitude (Quant) - CAT

Join OE and OD.
Internal angle of regular pentagon = 108°
∠EAB = ∠EDC = 108°
∠OAB = 60°
∠EAO = 48°
AO = OB = AB as the triangle is equilateral.
AB = AE as this is a regular pentagon.
Triangle AEO is isosceles as AO = EA.
∠AEO = ∠AOE = x (say)
In triangle AEO,
∠OAE + 2x = 180°
48° + 2x = 180°
2x = 132°, or x = 66° 
The question is " What is ∠OEA?" 
∠OEA = 66°
Hence, the answer is 66°
Choice A is the correct answer.

Question 9△ has sides a2, b2 and c2. Then the triangle with sides a, b, c has to be: 
A. Right-angled
B. Acute-angled
C. Obtuse-angled
D. Can be any of these three
Answer. (b)
Explanation.

Assume a ≤ b ≤ c,
we have a2 + b2 > c2.
This implies the triangle with sides a, b, c has to be acute-angled.
The question is "△ has sides a2, b2 and c2. Then the triangle with sides a, b, c has to be:"
Hence, the answer is Acute-angled .
Choice B is the correct answer.

Question 10Consider a right–angled triangle with inradius 2 cm and circumradius of 7 cm. What is the area of the triangle? 
A. 32 sq cms
B. 31.5 sq cms
C. 32.5 sq cms
D. 33 sq cms
Answer. (a)
Explanation.
r = 2
R = 7 (Half of hypotenuse)
Hypotenuse = 14
r = (a + b - h)/2
2 = (a + b - 14)/2
a + b - 14 = 4
a + b = 18
a2 + b2 = 14 2 
a2 + (18 - a)2 = 14 2 
a2 + 324 + a2 - 36a = 196
2a2 - 36a + 128 = 0
a2 - 18a + 64 = 0
Now the 2 roots to this equation will effectively be a, 18 – a. Product of the roots = 64.
Area = 1/2 * product of roots (How? Come on, you can figure this out.)
= 32 sq. cms
The question is "What is the area of the triangle?"
Hence, the answer is 32 sq. cms
Choice A is the correct answer.

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