# The hypotenuse of a right-angled triangle is larger than one of the legs by 4 cm and the second

The hypotenuse of a right-angled triangle is larger than one of the legs by 4 cm and the second by 18 cm. Find the perimeter of the triangle.

Consider a right-angled triangle with a hypotenuse, the length of which is greater than one of the legs by 4 cm, and the second by 18 cm. At the request of the task, we calculate the perimeter of this triangle.
The legs and hypotenuse of this right-angled triangle will be denoted by a, b and c, respectively. Then, according to the conditions of the assignment, c = a + 4 and c = b + 18 (hereinafter in the calculations, until the final result, we will omit the unit of measurement of length cm). Then, the legs of a right-angled triangle a and b are expressed through the hypotenuse with the following equalities a = c – 4 and b = c – 18.
We will use the Pythagorean theorem, the formula of which for our task can be formulated as the equality c² = a² + b² or, after substituting the expressions for a and b from the previous paragraph, c² = (c – 4) ² + (c – 18) ². Let’s use the abbreviated multiplication formula twice (a – b) ² = a² – 2 * a * b + b² (the square of the difference). Then, we have: c² = c² – 2 * c * 4 + 4² + c² – 2 * c * 18 + 18² or, giving similar terms on the right side, c² = 2 * c² – 24 * c + 340, whence c² – 44 * c + 340 = 0.
Let’s solve the resulting quadratic equation. It is easy to calculate that its discriminant D = (-44) ² – 4 * 1 * 340 = 1936 – 1360 = 576> 0. Therefore, the resulting quadratic equation has two different roots: c1 = (44 – √ (576)) / 2 = (44 – 24) / 2 = 10 and c2 = (44 + √ (576)) / 2 = (44 + 24) / 2 = 34. Let’s examine each root separately.
If c = 10, then the last formulas in item 2 allow you to calculate: a = c – 4 = 10 – 4 = 6 and b = c – 18 = 10 – 18 = -8, which is impossible for a triangle, that is, c = 10 – a side root.
For c = 34, the same formulas give the following true results: a = c – 4 = 34 – 4 = 30 and b = c – 18 = 34 – 18 = 16. It is easy to verify that a triangle with sides of 30 cm, 16 cm and 34 see exists.
Now we can easily calculate the required perimeter of the triangle: a + b + c = 30 cm + 16 cm + 34 cm = 80 cm.