Right triangles are one of the most common shapes you will encounter in math contests. Let’s do some practice problems:

• A triangle has a 90-degree angle. Its hypotenuse is 5 and one of its legs is 3. What is the length of the other leg?

The hypotenuse of a right triangle is the longer side that is opposite to the right angle. The legs are the two smaller sides. To solve this, we can use the Pythagorean Theorem.

The Pythagorean Theorem states that a^2 + b^2 = c^2. Here, a and b are the lengths of the two legs and c is the length of the hypotenuse.

Plugging in the values from the problem, we get that 9 + b^2 = 25. So, b^2 = 16 and b = 4.

Let’s do some problems involving special right triangles:

• A right triangle has a 45-degree angle and a 90-degree angle. One side is 4 units long and another side is larger than it. What is the area of the triangle?

The area of a triangle is base x height / 2. In a right triangle, the base can be equal to one of the legs and the height can be equal to another one of the legs. Thus, the area of a right triangle is the product of the two legs divided by two.

The sum of the angles in a triangle is 180. Since we know two of the angles, we can find the third one. The measure of the third angle is 180 - 90 - 45, which is 45. This means that this is an isosceles right triangle.

Isosceles right triangles have many unique properties. One of them is that the area is equal to one of the legs squared divided by two. We know that one of the side lengths in the triangle is 4, but how do we know if that is the hypotenuse?

Since the problem states that there is a side larger than 4, that side must be the hypotenuse. This is because the other leg must be 4, since this is an isosceles right triangle.

So, the area of the triangle is 42/2, which equals 8.

A 30-60-90 triangle is also a common special right triangle. In this type of right triangle, the side lengths follow this ratio:

1 : square root of 3 : 2.

The side opposite to the right angle is double the side opposite to the 30-degree angle. The side opposite to the 60-degree angle is equal to the other leg multiplied by the square root of three.

Right triangles can seem difficult and advanced, but they are prevalent and useful in many math contests.