A circle is inscribed in an isosceles triangle with a base of 12 cm and a perimeter of 32 cm. find the radius of this circle.

| August 14, 2021 | education

1. A, B, C – the vertices of the triangle. AC = 12 cm.

2. We calculate the lengths of the lateral sides AB and BC of the triangle:

AB = BC = (32 – 12) / 2 = 10 cm.

3. From the top B we draw the height BE. In an isosceles triangle, according to its properties, it also performs the functions of a median. The median divides the speaker side into two equal segments. Therefore, AE = CE = 12: 2 = 6 cm.

4. We calculate the length of the height BE. For the calculation, we use the Pythagorean theorem:

BE = √AB² – AE² = √10² – 6² = √100 – 36 = √64 = 8 cm.

5. Calculate the area (S) of a given triangle:

S = AC x BE / 2 = 12 x 8/2 = 48 cm².

6. Calculate the radius of the circle (r), which is inscribed in the triangle:

r = 2S / p.

p (perimeter) = 10 + 10 + 12 = 32 cm.

r = 2 x 48/32 = 3 cm.

Answer: the radius of the circle that is inscribed in the triangle is 3 cm.



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