Question

33) Высота прямоугольного треугольника, опущенная на гипотенузу равна корень из 10 , а один катет больше другого в 1,25 раза. Найти площадь треугольника. 2,5625

Answer 1

Решение на фото, удачи

Answer 2

Answer in detail. 33) The height of a right triangle, dropped to the hypotenuse, is equal to the square root of 10, and one leg is 1.25 times larger than the other. Find the area of the triangle. 2.5625

To solve this problem, we can use the properties of the height of a right triangle, which are explained in [this article](https://ege-study.ru/ru/ege/materialy/matematika/vysota-v-pryamougolnom-treugolnike-i-ee-svojstva/). Let us denote the legs of the triangle by a and b, and the hypotenuse by c. Let h be the height dropped to the hypotenuse. Then we have:

- h = sqrt(10) (given) - a = 1.25b (given) - c = sqrt(a^2 + b^2) (by Pythagoras' theorem) - S = ab/2 = ch/2 (the area of the triangle)

We need to find S, so we can substitute the given values and get:

S = ch/2 = sqrt(a^2 + b^2) * sqrt(10) / 2

Now we need to express a and b in terms of one variable. We can use the fact that a = 1.25b and get:

S = sqrt((1.25b)^2 + b^2) * sqrt(10) / 2 S = sqrt(2.5625b^2) * sqrt(10) / 2 S = 1.6b * sqrt(10) / 2

We can simplify this expression by dividing both sides by 0.8:

S / 0.8 = b * sqrt(10)

Now we can use the fact that S = 2.5625 (given) and solve for b:

2.5625 / 0.8 = b * sqrt(10) b = 0.8 * sqrt(10) / sqrt(10) b = 0.8

Finally, we can find a by using a = 1.25b:

a = 1.25 * 0.8 a = 1

Therefore, the legs of the triangle are a = 1 and b = 0.8, and the area of the triangle is S = 2.5625.

I hope this answer was helpful for you