# Solving Right Triangles

An arbitrary triangle is defined by three components, at least one of which must be a side. For example, a triangle can be defined by three sides \(\left({SSS}\right),\) or by a side and two adjacent angles \(\left({ASA}\right).\)

In a right triangle, one of the angles is equal to \(90^\circ.\) Therefore, to find all sides and angles of a right triangle, it is enough to know only two elements, at least one of which must be a side.

There are \(5\) basic combinations of sides and angles that uniquely define a right triangle:

- The hypotenuse and a leg \(\left({HL}\right),\)
- Two legs \(\left({LL}\right),\)
- The hypotenuse and an angle \(\left({HA}\right),\)
- A leg and the adjacent angle \(\left({LA_A}\right),\)
- A leg and the opposite angle \(\left({LA_O}\right),\)

Solving a triangle means finding all its sides and angles. Under finding an angle we mean finding a trigonometric function of the angle.

Depending on the given data, we use different relationships and identities. Let's consider these scenarios in more detail.

## 1. Solving a Right Triangle Given the Hypotenuse and a Leg \(\left({HL}\right)\)

Suppose the hypotenuse \(c\) and leg \(a\) are known for a right triangle.

To find the other leg \(b,\) we use the Pythagorean theorem:

Since all \(3\) sides are already known, we can determine any angle using a trigonometric function. For example,

The other acute angle can be expressed in terms of the sine or cosine as

## 2. Solving a Right Triangle Given Two Legs \(\left({LL}\right)\)

Let a right triangle be defined by legs \(a\) and \(b.\)

The hypotenuse \(c\) can be easily found by the Pythagorean theorem:

Now we are at the same point as in the previous case, so the acute angles \(\alpha\) and \(\beta\) can be defined in the same way:

## 3. Solving a Right Triangle Given the Hypotenuse and an Angle \(\left({HA}\right)\)

A right triangle has a hypotenuse \(c\) and an angle \(\alpha.\)

We find the legs of the triangle using the Pythagorean trig identity:

The other acute angle of the triangle is obviously

## 4. Solving a Right Triangle Given a Leg and the Adjacent Angle \(\left({LA_A}\right)\)

Consider a right triangle and suppose its leg \(b\) and the adjacent angle \(\alpha\) are known.

The hypotenuse can be found by the formula

The other leg \(a\) is given by

The angles \(\alpha\) and \(\beta\) are complementary, so

## 5. Solving a Right Triangle Given a Leg and the Opposite Angle \(\left({LA_O}\right)\)

The leg \(a\) and the opposite angle \(\alpha\) of a right triangle are known.

In this case, the hypotenuse can be found by the formula

To determine the other leg \(b,\) we can use the identity

The angle \(\beta\) is complementary to \(\alpha:\)

## General Case: Solving a Right Triangle Given Two Arbitrary Elements

In general, a right triangle can be defined by two arbitrary distinct elements. One of them must be a metric element (like a leg). For example, we could define a right triangle by its area and perimeter, or say, by an altitude drawn to the hypotenuse and a radius of the inscribed circle. Some pairs of such parameters lead to complex systems of equations. However, there is also a plenty of combinations of this type which are quite solvable. We will consider some of these problems below.