Question

Высота , проведенная из прямого угла треугольника, делит его гипотенузу на отрезки, равыне 4 и 9.Найдите площадь треугольника

Answer 1

Площадь треугольника находят половиной произведения высоты на основание. 
Основание данного треугольника ( гипотенуза) равна сумме отрезков, на которые делит ее высота. Пусть гипотенуза - с. Тогда 
с=4+9=13.
Высоту следует найти. 
Высота прямоугольного треугольника, проведенная из вершины прямого угла, есть среднее пропорциональное между отрезками, на которые делится гипотенуза этой высотой. ⇒
h²=4·9=36
h=√36=6
S=c·h:2=6·13:2=39 (ед. площади)

Answer 2

I can answer your question in detail. The height, drawn from the right angle of the triangle, divides its hypotenuse into segments equal to 4 and 9. Find the area of the triangle.

Let ABC be a right triangle with a right angle at C. Let CH be the height drawn from C to the hypotenuse AB. Then, by the property of the height of a right triangle, we have that CH divides AB into two segments AH and HB, such that AH = 4 and HB = 9. Also, the triangles ACH and BHC are similar to the triangle ABC, and the ratio of their sides is equal to the ratio of the corresponding segments of the hypotenuse.

To find the area of the triangle ABC, we can use the formula S = (1/2)ab, where a and b are the lengths of the legs. However, we do not know the values of a and b, so we need to find them using the given information. We can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. That is, a^2 + b^2 = c^2, where c is the length of the hypotenuse. In our case, c = AH + HB = 4 + 9 = 13. Therefore, we have:

a^2 + b^2 = 13^2 a^2 + b^2 = 169

Now, we can use the similarity of the triangles to find the values of a and b. Since the ratio of the sides of the triangles ACH and ABC is equal to the ratio of AH and AB, we have:

a / 4 = 13 / c a = 4 * 13 / c a = 52 / c

Similarly, since the ratio of the sides of the triangles BHC and ABC is equal to the ratio of HB and AB, we have:

b / 9 = 13 / c b = 9 * 13 / c b = 117 / c

Substituting these values into the equation a^2 + b^2 = 169, we get:

(52 / c)^2 + (117 / c)^2 = 169 (52^2 + 117^2) / c^2 = 169 (2704 + 13689) / c^2 = 169 16493 / c^2 = 169 c^2 = 16493 / 169 c^2 = 97.59

Taking the square root of both sides, we get:

c = sqrt(97.59) c = 9.88

This is the length of the hypotenuse AB. Now, we can find the lengths of the legs a and b by substituting c = 9.88 into the expressions we obtained earlier:

a = 52 / c a = 52 / 9.88 a = 5.26

b = 117 / c b = 117 / 9.88 b = 11.84

Finally, we can find the area of the triangle ABC by using the formula S = (1/2)ab:

S = (1/2) * 5.26 * 11.84 S = 2.63 * 11.84 S = 31.15

Therefore, the area of the triangle ABC is 31.15 square units.