Question

Сумма катетов прямоугольного треугольника равна 14 см гиппотенуза равна 10 см найдите катеты

Answer 1

5 и 9 сантиметров ................................

Answer 2

Solution for Finding the Length of the Catheti in a Right-Angled Triangle

To find the lengths of the catheti in a right-angled triangle, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Given: - Sum of the catheti (a and b): 14 cm - Length of the hypotenuse (c): 10 cm

We can use the Pythagorean theorem to find the lengths of the catheti.

Using the Pythagorean Theorem

According to the Pythagorean theorem, the sum of the squares of the lengths of the catheti is equal to the square of the length of the hypotenuse.

Let's denote: - Length of cathetus a: x - Length of cathetus b: y

The Pythagorean theorem can be expressed as: x^2 + y^2 = c^2

Substituting the given values: x^2 + y^2 = 10^2 x^2 + y^2 = 100

We also know that the sum of the catheti is 14 cm: x + y = 14

Solving for the Lengths of the Catheti

We have a system of equations: 1. x^2 + y^2 = 100 2. x + y = 14

We can solve this system of equations to find the lengths of the catheti.

Solution

Using the system of equations, we can solve for the lengths of the catheti: From equation (2), we can express y = 14 - x.

Substituting this into equation (1): x^2 + (14 - x)^2 = 100 x^2 + 196 - 28x + x^2 = 100 2x^2 - 28x + 96 = 0

Solving for x using the quadratic formula: x = (28 ± √(28^2 - 4*2*96)) / (2*2) x = (28 ± √(784 - 768)) / 4 x = (28 ± √16) / 4 x = (28 ± 4) / 4

So, we have two possible solutions for x: 1. x = (28 + 4) / 4 = 32 / 4 = 8 2. x = (28 - 4) / 4 = 24 / 4 = 6

Therefore, the lengths of the catheti are: - x = 8 cm - y = 6 cm

So, the lengths of the catheti in the right-angled triangle are 8 cm and 6 cm.