Question

У прямокутному трикутнику гіпотенуза на 4 см менша від суми катетів. Один з катетів дорівнює 8 см. Знайдіть сторони трикутника

Answer 1

A+b=c+4
a=8 см
a²+b²=c²
---
8+b=c+4
64+b²=c²
---
4+b=c
64+b²=(4+b)²
48-8b = 0
b = 6 см
c = 4+b = 10 см

Answer 2

Problem Analysis

We are given a right triangle where the hypotenuse is 4 cm less than the sum of the two legs. One of the legs is 8 cm long. We need to find the lengths of the sides of the triangle.

Solution

Let's assume the lengths of the two legs are a and b, and the length of the hypotenuse is c.

According to the problem statement, the hypotenuse is 4 cm less than the sum of the two legs. Mathematically, we can express this as:

c = a + b - 4 We are also given that one of the legs is 8 cm long. Let's assume this leg is a. Therefore, we have:

a = 8 Now, we can substitute the value of a from equation into equation to find the value of c:

c = 8 + b - 4 c = b + 4 We have two equations now: equation and equation We can solve these equations simultaneously to find the values of b and c.

Substituting the value of a from equation into equation we get:

c = 8 + b - 4 c = b + 4 [[4]]

Now, we can solve equations and [[4]] simultaneously:

a = 8 c = b + 4 [[4]]

Since we know that the sum of the angles in a triangle is 180 degrees, we can use this information to find the value of b.

Let's assume the right angle is A, the angle opposite to side a is B, and the angle opposite to side c is C.

We have:

A + B + C = 180 [[5]]

Since it is a right triangle, we know that angle A is 90 degrees. Therefore, we can rewrite equation [[5]] as:

90 + B + C = 180 B + C = 90 [[6]]

Now, we can use the fact that the sum of the angles in a triangle is 180 degrees to find the value of b.

In a right triangle, angle B is the complement of angle C. Therefore, we can rewrite equation [[6]] as:

B + (90 - B) = 90 2B = 90 B = 45 [[7]]

Now that we know the value of angle B, we can use trigonometric ratios to find the value of b.

In a right triangle, the sine of an angle is equal to the ratio of the length of the opposite side to the length of the hypotenuse. Therefore, we have:

sin(B) = b / c [[8]]

Substituting the values of B and c from equations [[7]] and [[4]], respectively, into equation [[8]], we get:

sin(45) = b / (b + 4)

Simplifying this equation, we can solve for b:

1 / sqrt(2) = b / (b + 4) b + 4 = b * sqrt(2) 4 = b * sqrt(2) - b 4 = b * (sqrt(2) - 1) b = 4 / (sqrt(2) - 1) [[9]]

Now that we have the value of b, we can substitute it back into equation [[4]] to find the value of c:

c = b + 4 [[4]]

Substituting the value of b from equation [[9]] into equation [[4]], we get:

c = (4 / (sqrt(2) - 1)) + 4

Simplifying this equation, we can solve for c:

c = (4 + 4(sqrt(2) - 1)) / (sqrt(2) - 1) [[10]]

Therefore, the lengths of the sides of the triangle are: - a = 8 cm - b = 4 / (sqrt(2) - 1) cm (approximately 6.83 cm) - c = (4 + 4(sqrt(2) - 1)) / (sqrt(2) - 1) cm (approximately 14.14 cm)

Answer

The lengths of the sides of the triangle are approximately: - a = 8 cm - b = 6.83 cm - c = 14.14 cm