In the adjoining figure, △ A B C is right-angled at B . If B C = 6 cm and A B = 8 cm , find the radius of the incircle.

Given, B C = 6 cm A B = 8 cm We know that in right angle triangle: A C 2 = A B 2 + B C 2 A C 2 = 8 2 + 6 2 A C 2 = 64 + 36 A C 2 = 100 A C = 10 cm Now Area of △ A B C = Area of △ O A B + Area of △ O B C + Area of △ O C A ⇒ 1 2 × B C × A B = 1 2 × O D × A B + 1 2 × O E × B C + 1 2 × O F × A C ⇒ 1 2 × 6 × 8 = 1 2 × r × 8 + 1 2 × r × 6 + 1 2 × r × 10 ⇒ 24 = 12 r ⇒ r = 2 Thus the radius of the circle is 2 cm .

In the figure, a , b , c are the sides of a right triangle where c is the hypotenuse. Prove that the radius r of the circle which touches the sides of the triangle is given by r = a + b + c 2 .

Given, a , b , c are the sides of a right triangle where c is the hypotenuse. To prove: r = a + b + c 2 Proof: Given, ∠ C = 90 ° So, ∠ D O E C is square whose each side is equal to radius of circle, r . Now, B F = B D = B C - C D = a - r And A F = A E = A C - E C = b - r ∴ A B = B F + A F = a - r + b - r ⇒ c = a + b - 2 r ⇒ 2 r = a + b - c ⇒ r = a + b - c 2 Hence, proved.

E M B I B E

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 10 - Circles>Nagaland Board-2020>Q 9

HARD

10th Nagaland Board

IMPORTANT

Earn 100

Prove that the parallelogram circumscribing a circle is a rhombus.

Important Questions on Circles

EASY

10th Nagaland Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 10 - Circles>Nagaland Board-2018>Q 2

In the adjoining figure, $\Rightarrow AB = BC$ and $P B$ are two tangents to the circle with centre $AB = BC = CD = AD$ , such that $∠ A O B = 110 °$ , then $∠ A P B$ is

Question Image

EASY

10th Nagaland Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 10 - Circles>Nagaland Board-2018>Q 3

Two tangents

T P

and

T Q

are drawn to a circle with centre

O

from an external point

T

. Prove that

∠ P T Q = 2 ∠ O P Q

.

HARD

10th Nagaland Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 10 - Circles>Nagaland Board-2019>Q 7

Question Image

In the adjoining figure, $P Q, R S$ and $X Y$ are tangent at $A, B$ and $Z$ respectively. $A O B$ is the diameter of the circle. If $∠ A X O = 25 °$ , find $∠ B O X$ .