In a right-angled triangle ABC, the height CP is drawn so that the length of the segment AP is 4 cm longer than the length

| September 9, 2021 | education

In a right-angled triangle ABC, the height CP is drawn so that the length of the segment AP is 4 cm longer than the length of the segment CP, and BP = 9 cm. Find the hypotenuse of triangle ABC.

The height СР divides the triangle ABC into two similar in acute angle triangle BCP and ACP.

Then CP / AP = BP / CP.

Let the length of the height CP = X cm, then, by condition, AP = X + 4 cm.

X / (X + 4) = 9 / X.

X ^ 2 = 9 * X + 36.

X ^ 2 – 9 * X – 36 = 0.

Let’s solve the quadratic equation.

D = b^2 – 4 * a * c = (-9) ^ 2 – 4 * 1 * (-36) = 81 + 144 = 225.

X1 = (9 – √225) / 2 * 1 = (9 – 15) / 2 = -6 / 2 = -3. (Doesn’t fit.)

X2 = (9 + √225) / 2 * 1 = (9 + 15) / 2 = 24/2 = 12.

CP = 12 cm, then AP = X + 4 = 12 + 4 = 16 cm.

AB = AP + BP = 16 + 9 = 25 cm.

Answer: The length of the hypotenuse is 25 cm.



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