In a rectangular triangle CBA, the height CK (from the right angle) is 12 cm, the hypotenuse AB is equal to 25 cm.

| September 4, 2021 | education

In a rectangular triangle CBA, the height CK (from the right angle) is 12 cm, the hypotenuse AB is equal to 25 cm. Find KB (the segment by which the height divides the AB side).

The height CK divides triangle ABC into two right-angled triangles, ACK and BCK.

Let us prove the similarity of triangles ACK and BCK.

Let the angle CAK = X0, then the angle ACK = (90 – X) 0. Angle АСВ = 90, then angle ВСК = (90 – (90 – X)) = X0.

Then the right-angled triangle ACK is similar to the right-angled triangle BCK in an acute angle.

In such triangles, the ratio of similar sides is equal, then: AK / СK = СK / ВK.

CK ^ 2 = AK * ВK.

Let the length of ВK = X cm, then AK = (25 – X).

Then 144 = (25 – X) * X.

X ^ 2 – 25 * X + 144 = 0.

Let’s solve the quadratic equation.

X1 = 9 cm.

X2 = 16 cm.

Answer: The length of the ВC segment is 9 cm or 16 cm.



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