The height CD of a right-angled triangle ABC cuts off a segment equal to 10 cm from the hypotenuse AB, AD = 4 cm

| August 3, 2021 | education

The height CD of a right-angled triangle ABC cuts off a segment equal to 10 cm from the hypotenuse AB, AD = 4 cm, to prove that the ABC triangle is similar to the triangle ACD, find AC.

Determine the length of the segment BD. BD = AB – AD = 10 – 4 = 6 cm.

In a right-angled triangle, the square of the height drawn from the vertex of the right angle is equal to the product of the line segments by which it divides the hypotenuse.

CD ^ 2 = AD * BD = 4 * 6 = 24 cm.

СD = √24 cm.

From the right-angled triangle ACD we define the hypotenuse AC.

AC ^ 2 = AD ^ 2 + CD ^ 2 = 16 + 24 = 40.

AC = √40 cm.

Let us prove the similarity of triangles ABC and ACD.

AC / AB = AD / AC.

AC ^ 2 = AB * AD.

(√40) 2 = 10 * 4.

40 = 40. Since the proportion is correct, the triangles are similar.

Answer: Length AC = √40 cm.



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