The hypotinus of a right-angled triangle is 15 cm, and the area of the triangle is 54 cm ^ 2. Find the legs of a right-angled triangle.

| January 15, 2021 | education

Let’s designate the legs of a right-angled triangle A, B, and the hypotenuse – C.
C = 15 cm, S = 54 cm2.
The area of ​​a right-angled triangle is determined by the formula S = ½ * A * B.
The hypotenuse in a right-angled triangle can also be found by the formula: C ^ 2 = A ^ 2 + B ^ 2.
We get the following equations:
A ^ 2 + B ^ 2 = 255.
AB = 108.
We use the formula for the square of the sum: A ^ 2 + 2AB + B ^ 2 = (A + B) ^ 2.
(A + B) ^ 2 = 225 + 2 * 108.
(A + B) ^ 2 = 441.
A + B = 21.
Now we use the formula for the square of the difference: A ^ 2 – 2AB + B ^ 2 = (A – B) ^ 2.
(A – B) ^ 2 = 225 – 2 * 108.
(A – B) ^ 2 = 9.
A – B = 3.
Received 2 linear equations A + B = 21 and A – B = 3.
Let’s add them up and get:
A + B + A – B = 24.
2A = 24.
A = 12.
We calculate the value of B.
B = 21 – A.
B = 21 – 12.
B = 9.
Answer: the legs of the triangle are 12 and 9.



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