One of the legs of a right-angled triangle is 12 cm, and the hypotenuse is 13 cm. Find the second leg

One of the legs of a right-angled triangle is 12 cm, and the hypotenuse is 13 cm. Find the second leg and the hypotenuse of the triangle.

Brief notation of the problem condition
Let us formulate the given condition of the problem in the form of a short notation.

Given:

△ ABC;

∠С = 90 °;

AB = 13 cm;

AC = 12 cm;

BC =?

Defining a Right Triangle
A triangle is called rectangular if it has a right angle, i.e. an angle of 90 degrees. The side of a right-angled triangle opposite the right angle is called the hypotenuse, the other two sides are called legs.

In this problem, angle C is a straight line, AB is the hypotenuse, AC and BC are legs.

Pythagorean theorem

To find the leg of a right-angled triangle, apply the Pythagorean theorem, according to which in a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the legs.

It follows from the Pythagorean theorem that:

AB ^ 2 = AC ^ 2 + BC ^ 2;

Finding the leg
Find the leg of a right-angled triangle, sequentially performing the following steps:

based on the Pythagorean theorem, the square of the leg is equal to the difference between the squares of the hypotenuse and the known leg;
leg is equal to the square root of the difference between the squares of the hypotenuse and the known leg;
substitute the given values ​​of the hypotenuse and leg into the resulting formula;
calculate the value of the square of the hypotenuse and the known leg: 13 ^ 2 = 169; 12 ^ 2 = 144;
find the difference between the squares of the hypotenuse and the known leg: 169 – 144 = 25;
calculate the square root of the difference between the squares of the hypotenuse and the leg: √25 = 5;
BC ^ 2 = AB ^ 2 – AC ^ 2;

BC = √AB ^ 2 – AC ^ 2;

BC = √13 ^ 2 – 12 ^ 2;

BC = √169 – 144;

BC = √25;

BC = 5 (cm).

Therefore, the leg BC of a right-angled triangle ABC is 5 cm.

Answer: 5 cm.