In a rectangle ABC with a right angle c, the leg AC = 12 and the hypotenuse AB = 13

| June 25, 2021 | education

In a rectangle ABC with a right angle c, the leg AC = 12 and the hypotenuse AB = 13 are known, find the radius of the inscribed circle.

According to the condition of the problem, in this right-angled triangle ABC, the length of the AC leg is 12, and the length of the hypotenuse AB is 13.

Using the Pythagorean theorem, we find the length of the second leg of the BC:

| Sun | = √ (| AB | ^ 2 – | AC | ^ 2) = √ (13 ^ 2 – 12 ^ 2) = √ (169 – 144) = √25 = 5.

Knowing the two legs of a given right-angled triangle, we find its area S:

S = | AB | * | Sun | / 2 = 12 * 5/2 = 6 * 5 = 30.

Knowing what the three sides of a given triangle are equal to, as well as its area, we can find the radius r of the circle inscribed in this triangle using the formula:

S = r * (| AB | + | BC | + | AC |) / 2.

Find the radius r:

r = S / ((| AB | + | BC | + | AC |) / 2) = 30 / ((12 + 5 + 13) / 2) = 30 / (30/2) = 30/15 = 2.

Answer: the radius of the inscribed triangle is 2.



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