ABC is a right angle triangle right angled at A .a circle is inscribed in it the length of the two sides containing the right angles are 6 cm and 8 cm. find the radius of the circle?

Question

Answer

Answer

Answer

Solution:

Circumradius of a Triangle:

For any triangle with sides a, b, and c, the circumradius (R) can be found using the formula:

R = abc / 4Δ, where Δ is the area of the triangle.

Incircle of a Triangle:

The incircle of a triangle is a circle that is tangent to all three sides of the triangle.

The radius of the incircle is given by the formula:

r = Δ / s, where s is the semiperimeter of the triangle (i.e., s = (a + b + c) / 2).

Given:

ABC is a right angle triangle, right angled at A.

The length of the two sides containing the right angles are 6 cm and 8 cm.

Approach:

First, we need to find the length of the hypotenuse of the triangle using the Pythagorean theorem.

Then, we can find the semiperimeter of the triangle, s.

Using the formula for the radius of the incircle, we can find the radius of the circle.

Solution:

Using the Pythagorean theorem, we can find the length of the hypotenuse of the triangle as:

c^2 = a^2 + b^2 = 6^2 + 8^2 = 100

c = √100 = 10 cm

The semiperimeter of the triangle is:

s = (a + b + c) / 2 = (6 + 8 + 10) / 2 = 12 cm

Using the formula for the radius of the incircle, we can find the radius of the circle as:

r = Δ / s, where Δ is the area of the triangle.

Since the triangle is right angled at A, we can use the formula Δ = ab / 2, where a and b are the lengths of the legs of the triangle.

Δ = ab / 2 = 6 * 8 / 2 = 24 cm^2

Therefore, the radius of the circle is:

r = Δ / s = 24 / 12 = 2 cm

Answer:

The radius of the inscribed circle is 2 cm.

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