The height drawn from the top of the right angle of a right-angled triangle is 6 and divides the hypotenuse

| July 29, 2021 | education

The height drawn from the top of the right angle of a right-angled triangle is 6 and divides the hypotenuse into segments, one of which is 5 cm larger than the other. Find: The ratio in which the given height divides the area of the triangle.

Let the length of the segment BH = X cm, then the length of the segment AH = (X + 5) cm.

Let us prove the similarity of triangles ACH and BCH.

Let the value of the angle HAC of the triangle ABC be equal to X0, then the angle ACH = (90 – X) 0.

Angle АСВ = 90, then angle ВСН = (90 – (90 – X) = X0.

The acute angles of the right-angled triangles АСН and ВСН are equal, then the triangles are similar in acute angle.

Then in similar triangles АН / СН = СН / ВН.

CH ^ 2 = AH * BH.

36 = (X + 5) * X.

X2 + 5 * X – 36 = 0.

Let’s solve the quadratic equation.

X = BH = 4 cm.

Then AH = 4 + 5 = 9 cm.

Since the height of CH is common for triangles ACH and BCH, the ratio of their areas is equal to the ratio of the lengths of the bases.

Svsn / Ssn = BH / AH = 5/9.

Answer: The area of ​​a triangle is divided by the ratio of 5/9.



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