A right-angled triangle with a perimeter of 15 is inscribed with a circle of radius 1. Find the sides of this triangle.
| March 19, 2021 | education
The radius of a circle inscribed in a right-angled triangle is:
R = 1 = (AB + AC – BC) / 2.
(AB + AC – BC) = 2 cm.
By condition, AB + AC + BC = 15 cm.
Let us subtract the first equality from the second.
(AB + AC + BC) – (AB + AC – BC) = 15 – 2.
2 * BC = 13.
BC = 13/2 = 6.5 cm.
Then AB + AC = 2 + 6.5 = 8.5 cm.
AB = 8.5 – AC.
By the Pythagorean theorem, BC ^ 2 = AC ^ 2 + AB ^ 2 = 6.5 ^ 2.
42.25 = AC ^ 2 + (8.5 – AC) ^ 2 = AC ^ 2 + 72.25 – 17 * AC + AC ^ 2.
2 * A ^ C2 – 17 * AC + 30 = 0.
Let’s solve the quadratic equation.
AC1 = 2.5 cm.
AC2 = 6 cm.
If AC = 2.5 cm, then AB = 8.5 – 2.5 = 6 cm.
If AC = 6 cm, then AC = 8.5 – 6 = 2.5 cm.
Answer: The sides of the triangle are 2.5 cm, 6 cm, 6.5 cm.
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