Question

В треугольнике ABC высота CH, опущенная из вершины прямого угла C делит гипотенузу AB на отрезки AH=4 см и BH=4 см. Чему равен катет BC? Прошу полное решение

Answer 1

Треугольники АНС = СНВ, СН общая, АН=НВ, углы АНС=СНВ=90

соответственно их углы и стороны равны, АС=АВ

по т. Пифагора AC^2 + BC^2 = AB^2

AB=8, AC=BC=x

x^2 + x^2 = 8^2 =64

2*x^2 =64

x^2 = 32

x = √32 =√(16*2) = 4√2

т.о. катет ВС = 4√2 см

Ответ: 4√2 см


 

Answer 2

Problem Analysis

We are given a right triangle ABC, with the altitude CH dropped from the vertex of the right angle C. The altitude divides the hypotenuse AB into two segments, AH and BH, with lengths 4 cm each. We need to find the length of the leg BC.

Solution

To solve this problem, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Let's denote the length of BC as x. We can set up the following equation using the Pythagorean theorem:

AC^2 + BC^2 = AB^2

Since AC is the length of the leg opposite to angle C, and AB is the hypotenuse, we can substitute the given values:

5^2 + x^2 = (AH + BH)^2

Substituting the given values AH = 4 cm and BH = 4 cm:

5^2 + x^2 = (4 + 4)^2

Simplifying the equation:

25 + x^2 = 64

Subtracting 25 from both sides:

x^2 = 39

Taking the square root of both sides:

x = √39

Therefore, the length of the leg BC is approximately √39 cm.

Answer

The length of the leg BC in the triangle ABC is approximately √39 cm.