The height of a right-angled triangle, drawn to the hypotenuse, divides it into segments of 9 cm and 16 cm

| August 8, 2021 | education

The height of a right-angled triangle, drawn to the hypotenuse, divides it into segments of 9 cm and 16 cm. Find the radius of the circle inscribed in the triangle.

Let a right-angled triangle ABC be given, where ∠BCA = 90º. The height of the CК, drawn to the hypotenuse ВA, divides it into segments ВK = 9 cm and CA = 16 cm.Then:

9 + 16 = 25 (cm) – length of the hypotenuse ВA, since ВA = ВK + KA;

√ (9 · 25) = 15 (cm) – the size of the BC leg, since according to the property of the height drawn to the hypotenuse, BC² = ВK · VA;

√ (16 · 25) = 20 (cm) – the size of the AC leg, since by the property of the height drawn to the hypotenuse, AC² = AK · ВA;

25 + 15 + 20 = 60 (cm) – the perimeter of the triangle p;

(15 20): 2 = 150 (cm²) – the area of ​​the triangle S.

To find the radius of a circle r inscribed in a triangle, we use the formula: r = 2 · S / p.

Substitute the found values ​​of the quantities into the formula and find the radius:

r = 2 * 150/60;

r = 5 cm.

Answer: the radius of a circle r inscribed in a triangle is 5 centimeters.



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