Question

a triangle abc is circumscribe a circle ,AB equal to 12 centimetre BC is equal to 8 cm is equal to 10 cm find the radius of the circle​

Answer 1

Answer: radius of the triangle = 2 cm

Step-by-step explanation:

In a right angle triangle, in-circle touches all the inner sides, means the small two sides are the tangents to the circle.

In above image, O is centre of in-circle.

BO = √2*r

OM = r. BO and OM are collinear or forms a single line. Think about it?

So BM = (√2 + 1) * r

Now BM is the altitude of the triangle.

Area if the triangle, since it is right angled triangle,

Area = (1/2)* (altitude)*(hypotenuse)

hypotenuse = 10 cm

(1/2)*(√2+1)*r * 10 = (1/2)*(6)*(8)

(√2 +1)*r = 4.8

r = (4.8)*(√2–1)

For any right angled triangle , the radius of in-circle

r = {(a*b)/c}*(√2 - 1)

where a and b are the side lengths and c is hypotenuse length

Since ABC is a right angled triangle And we are given BC and AB ..hence using Pythagoras theorem we can easily find AC

AB2+BC2=AC2AB2+BC2=AC2

therefore, AC = 10 cm

Now, we draw the in circle for this triangle as follows…

Here, I have used the concept that the lengths of tangents drawn to circle from any point outside it is equal…

So, we get AC= 8-x+6-x

i.e. AC = 14 - 2x

But Since AC was 10 cm

So, 14 - 2x = 10

2x = 4

x=2 cm

So, the in radius of the triangle ABC = 2 cm.

Hope it helps you!!!