Triangle ABC. Angle C = 90 ° BC = 6 cm, the projection of the leg AC on the hypotenuse is 5 cm

| September 18, 2021 | education

Triangle ABC. Angle C = 90 ° BC = 6 cm, the projection of the leg AC on the hypotenuse is 5 cm. Find the area of the triangle ABC.

Let’s construct the perpendicular CH. Then AН is the projection of the AC leg onto the hypotenuse AB.

By the property of the leg of a right-angled triangle: the leg of a right-angled triangle is the geometric mean between the hypotenuse and the projection of this leg onto the hypotenuse.

Then BC ^ 2 = AB * BН.

The length of the hypotenuse AB = BH + AH = BH + 5 cm.

6 ^ 2 = (BH + 5) * BH.

BH ^ 2 + 5 * BH – 36 = 0.

Let’s solve the quadratic equation.

BH = 4 cm, then AB = 4 + 5 = 9 cm.

By the Pythagorean theorem, AC ^ 2 = AB ^ 2 – BC ^ 2.

AC ^ 2 = 81 – 36 = 45.

AC = 3 * √5.

Determine the area of ​​the triangle ABC.

Sас = ВС * АС / 2 = 6 * 3 * √5 / 2 = 9 * √5 cm ^ 2.

Answer: The area of ​​the triangle is 9 * √5 cm ^ 2.



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