The radius of a circle inscribed in a right-angled triangle is 2 cm and the sum of the legs is 17

The radius of a circle inscribed in a right-angled triangle is 2 cm and the sum of the legs is 17 cm, find the perimeter of the triangle and its area.

ABC – right triangle, angle C = 90 degrees, AB = c – hypotenuse, BC = a and AC = b – legs (by condition a + b = 17 cm), r = 2 cm – radius of a circle inscribed in a right triangle.
1. The radius of a circle inscribed in a right-angled triangle is found by the formula:
r = (a + b – c) / 2.
Substitute the known values ​​and find the length of the hypotenuse with:
(17 – c) / 2 = 2;
17 – c = 4;
– c = – 13;
c = 13 cm.
The perimeter of the triangle ABC is:
P = a + b + c;
P = 17 + 13 = 30 (cm).
2. The area of ​​a right-angled triangle is equal to half the product of its legs:
S = ab / 2.
Let’s find the lengths of the legs ABC. Let’s compose a system of equations:
a + b = 17;
a ^ 2 + b ^ 2 = 169 (by the Pythagorean theorem).
In the first equation, we express a through b:
a = 17 – b.
Substitute the resulting expression into the second equation and find the length b:
(17 – b) ^ 2 + b ^ 2 = 169;
289 – 34b + b ^ 2 + b ^ 2 = 169;
2b ^ 2 – 34b + 120 = 0;
b ^ 2 – 17b + 60 = 0.
Discriminant:
D = 17 ^ 2 – 4 * 60 = 289 – 240 = 49.
b1 = (17 – 7) / 2 = 10/2 = 5 (cm).
b2 = (17 + 7) / 2 = 24/2 = 12 (cm).
Then: a1 = 17 – b1 = 17 – 5 = 12 (cm);
a2 = 17 – b2 = 17 – 12 = 5 (cm).
The ABC area is equal to:
S = 5 * 12/2 = 60/2 = 30 (cm ^ 2).
Answer: P = 30 cm, S = 30 cm ^ 2




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