Competitive Exam PYQ

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Q. What is the area of the incircle of a right-angled triangle having sides 3 cm, 4 cm and 5 cm ?

A. π cm²  B. 4π cm²  C. 2π cm²  D. 3π cm²

Solution

The sides of the right-angled triangle are: Base = 3 cm, Height = 4 cm, Hypotenuse = 5 cm

Formula for the inradius: r = a + b − c/2​

                                         r = 3 + 4 − 5​/2 = 2/2​ = 1 cm

Now, area of the incircle: $$\text{A = }$$πr²

$$\text{ A = }$$π(1)²

$$\text{ A = π }$$cm²

^{Ans: A. π cm²}

​Trick

For a 3 – 4 – 5 right triangle:

$$\text{r = 3 + 4 − 5/2 = 1 cm}$$

So, Incircle Area = π cm².

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Q. What is the area of the incircle of a right-angled triangle having sides 8 cm, 15 cm, and 17 cm ?

A. 4π cm²  B. 9π cm²  C. 16π cm²  D. 25π cm²

Solution

For a right-angled triangle: r = a + b − c/2

Here,

• $a=8$ cm, $b=15$ cm, $c=17$ cm

r = 8 + 15 − 17/2​

r = 6​/2 = 3 cm

Now, area of the incircle:  A = πr²

A = π(3)² = 9π cm²

Ans: B. 9π cm²

​Trick

For an 8–15–17 right triangle:

r = 8+15−17​/2 = 3 cm​

Therefore,

Incircle Area = 9π cm²

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Q. What is the area of the circumcircle of a right-angled triangle having sides 5 cm, 12 cm, and 13 cm ?

A. 36π cm²  B. 42π cm²  C. 42.25π cm²  D. 28.25π cm²

Solution

For a right-angled triangle, the hypotenuse is the diameter of the circumcircle.

Diameter = 13 cm

Therefore,

R = 13/2 ​= 6.5 cm

Now, area of the circumcircle: A = πR² = π(6.5)² = 42.25π cm²

Ans: C. 42.25π cm²

Trick

For a right triangle: R = Hypotenuse/2​​

So,

R = 13/2 ​= 6.5

Circumcircle Area = 42.25π cm²

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Q. What is the area of the circumcircle of a right-angled triangle having sides 6 cm, 8 cm, and 10 cm ?

A. 16π cm²  B. 18π cm²  C. 20π cm²  D. 25π cm²

Solution

For a right-angled triangle, the hypotenuse is the diameter of the circumcircle.

Here, H=10 cm

Therefore, R = H​/2 = 10/2 ​= 5 cm

Now, area of the circumcircle:  A = πR² = π(5)² = 25π cm²​

​Ans: D. 25π cm²

​Trick

For a right triangle: R = Hypotenuse / 2​​

So,

R = 10​/2 = 5 Circumcircle Area=25π cm²

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Polygon – Interior Angle

Q1.If each interior angle of a regular polygon is 160°, find the number of sides.

A) 18  B) 15  C) 17  D) 14

Formula:

n = 360/180−120 = 360/60 = 6

Ans: A) 18

Q2.If each interior angle of a regular polygon is 120°, find the number of sides.

A) 10  B) 8  C) 6  D) 5

Solution:

n = 360/180−120 = 360/60 = 6

Ans: C) 6

Short Trick

Number of sides = 360 ÷ (180 − Interior Angle)

For 120°:

360 ÷ 60 = 6

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