The bisector of an acute angle of a right-angled triangle divides the leg into segments, one of which is 2 cm less than the other.

The bisector of an acute angle of a right-angled triangle divides the leg into segments, one of which is 2 cm less than the other. Find the area of a triangle if the hypotenuse and the second leg are 5: 4

Let the length of the hypotenuse be equal to AB = 5 * X cm, then, by condition, the leg BC = 4 * X cm.

By the Pythagorean theorem, AC ^ 2 = AB ^ 2 – BC ^ 2 = 25 * X ^ 2 – 16 * X ^ 2 = 9 * X2.

AC = 3 * X cm.

Let the length of the segment AM = Y cm, then CM = (Y – 2) cm.

By the property of the angle bisector:

AB / AM = BC / CM.

5 * X / Y = 4 * X / (Y – 2).

5 * Y – 10 = 4 * Y.

Y = 10.

AC = 10 + (10 – 2) = 18 cm.

18 = 3 * X.

X = 18/3 = 6 cm.

Then BC = 4 * 6 = 24 cm.

Savs = AC * BC / 2 = 18 * 24/2 = 216 cm2.

Answer: The area of the triangle is 216 cm2.




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