In a right-angled triangle, the hypotenuse is 13, and the height drawn from the vertex of the right angle is 6.

| March 9, 2021 | education

In a right-angled triangle, the hypotenuse is 13, and the height drawn from the vertex of the right angle is 6. Find the sides of the triangle.

By the property of the height of a right-angled triangle, the square of its length is equal to the product of the segments into which the height divides the hypotenuse.

AH ^ 2 = BH * CH.

Let CH = X cm, then BH = (13 – X) cm.

Then 36 = (13 – X) * X = 13 * X – X ^ 2.

X ^ 2 – 13 * X + 36 = 0.

Let’s solve the quadratic equation.

D = b ^ 2 – 4 * a * c = (-13) ^ 2 – 4 * 1 * 36 = 169 – 144 = 25.

X1 = (13 – √25) / (2 * 1) = (13 – 5) / 2 = 8/2 = 4.

X2 = (13 + √25) / (2 * 1) = (13 + 5) / 2 = 18/2 = 9.

CH = 4 cm, then BH = 9 cm.

From the right-angled triangle AНС, by the Pythagorean theorem, we determine the length of the AС.

AC ^ 2 = AH ^ 2 + CH ^ 2 = 36 + 16 = 52. AC = √52 = 2 * √13 cm.

From the right-angled triangle AНВ, by the Pythagorean theorem, we determine the length AB.

AB ^ 2 = BH ^ 2 + AH ^ 2 = 81 + 36 = 117. AB = √117 = 3 * √13 cm.

Answer: The sides of the triangle are 2 * √13 cm, 3 * √13 cm, 13 cm.



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