How to Solve Right Triangles?

In this article, we’ll learn about how to Solve Right Triangles. But first, learn about the Triangles. 

Triangles are made up of three line segments. These three segments meet to form three angles. The lengths of the sides and sizes of the angles are related to one another. If you know the size (length) of three out of the six parts of the triangle (at least one side must be included), you can find the sizes of the remaining sides and angles.

And if the triangle is right-angled, you can use simple trigonometric ratios to find the missing parts. In a general triangle that is acute or obtuse, you will need to use other techniques like the sine and cosine laws. 

How to Solve a Right Triangle?

You should know the methodology to solve right triangles. Let the three angles of a triangle ABC be labeled as ∠ A, ∠ B, and ∠ C in capital letters, and the three sides of the triangle should be labeled as a, B, and C in small letters. The diagram below shows the labeled figure. If any three of these six measurements are known (other than knowing the measures of the three angles), you can calculate the values of the other three measurements. 

Solving right triangles here means finding the missing values in the triangle. One of the angles is 90° if the triangle is a right triangle. Therefore, you can solve the right triangle if you are given the measures of two of the three sides or even when you are given the measure of one side and one of the other two angles.

A right triangle is a triangle in which one angle is a right angle that is an angle with a value of 90 degrees (90°). The relation between the sides and angles of a right triangle is the basis for trigonometry.

The side opposite the right angle is called the hypotenuse. It is side c in the above figure. The sides adjacent to the right angle are called the legs of the triangle. Here sides A and B are the legs. Side A is the side adjacent to angle B and opposite to angle A. Side B is the side adjacent to angle A and opposite to angle B.

In several applications, specific angles are given unique names. The Angle of elevation and the Angle of depression are two of these unique terms. 

The Pythagorean Theorem

The Pythagorean Theorem, also known as Pythagoras’ Theorem, helps to define the relationship among the three sides of a right triangle. Pythagoras’ theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as the below-given equation where a, b, and c are the three sides of the triangle ABC.

a²+b²=c²

In this equation, c represents the length of the hypotenuse, and a and b are the lengths of the triangle’s other two sides.

Example:  

A right triangle has side lengths of 3 cm and 4 cm.  Find the length of the hypotenuse.

Substitute 

a=3 and b=4 into the Pythagorean Theorem and solve for c

a²+b²=c²

3²+4²=c²

9+16=c²

25=c²

c²=25

√c²=√25

c=5 cm

Solving for a Side in Right Triangles with the Help of Trigonometry Functions

When we say to solve a triangle, we mean to know all three sides and all three angles. Let us look at finding the value of a missing side length in a right triangle by choosing the right trigonometric ratio for a given angle.

When working with right triangles and circles, the sine, cosine, and tangent ratios are three of the most fundamental tools. Let us understand this further.

The sum of internal angles in a triangle is 180°. If we have a right triangle with a non-right angle of 𝜃, the remaining angle will measure 90°−𝜃.

Then, we recall the AAA criterion for similarity. This tells us that two triangles are similar if they have the same internal angles. If two triangles are similar, then the lengths of their sides are scalar multiples of each other.

These two facts allow us to notice an interesting property of right triangles: the value of the ratio of any two lengths of the sides of a right triangle is only dependent on the angle and the choice of the two sides.

We can use trigonometric ratios to find unknown sides in right triangles.

Let’s look at an example.

Given △ABC, A, B, and C, find AC

Solution

Step 1: Determine which trigonometric ratio to use.

Let’s focus on angle B since that is the angle that is explicitly given in the diagram.

Note that we are given the length of the hypotenuse, and we are asked to find the length of the side opposite angle B. The trigonometric ratio that contains both of those sides is the sine. 

Step 2: Create an equation using the trig ratio sine and solve for the unknown side.

sin(B) =  opposite / hypotenuse (Define sine)

sin(50°) = AC/6 (Substitute)

6sin(50°) = AC

4.60 ≈ AC                         ​

Finding an Angle in a Right-Angled Triangle

If we know the lengths of two of its sides, we can find an unknown angle in a right-angled triangle.

This can be easily done by Sine, Cosine, or Tangent!

A special phrase to remember which one to use is ”OLD HARRY AND HIS OLD AUNT”

O – Opposite

H – Hypotenuse

A –  Adjacent

Sine: sin(θ) = Opposite / Hypotenuse.

    Cosine: cos(θ) = Adjacent / Hypotenuse

Tangent: tan(θ) = Opposite / Adjacent

Example:

Find AB

Tan x° = opposite/adjacent = 300/400 = 0.75

Therefore, tan-1 of 0.75 = 36.9°

How do Solve Special Right Triangles?

To solve for a side in right triangles, first, you should recognize a right-angled triangle. A special right triangle is a right triangle whose sides are in a particular ratio, called the Pythagorean triplets. You can also use the Pythagoras’ theorem”, but if you can see that it is a special triangle, it can save you some calculations.

What is a 45°-45°-90° Triangle?

A 45°-45°-90° triangle is a special right triangle whose angles are 45°, 45°, and 90°. The lengths of the sides of a 45°-45°-90° triangle are in the ratio of 1: 1: √2.

A right triangle with two sides of equal lengths must be a 45°-45°-90° triangle.

You can also recognize a 45°-45°-90° triangle by the angles. A right triangle with a 45° angle must be a 45°-45°-90° special right triangle.

Side1 : Side2 : Hypotenuse = x : x : x√2

Example 1:

Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are both 3 inches.

Solution:

Step 1: This is a right triangle with two equal sides so it must be a 45°-45°-90° triangle.

Step 2: You are given that both sides are 3. If the first and second value of the ratio x:x:x√2 is 3 then the length of the third side is 3√2.

Answer: The length of the hypotenuse is 3√2 inches.

What is a 30°-60°-90° Triangle?

Another special right triangle is the 30°-60°-90° triangle. This is a right triangle whose angles are 30°-60°-90°. The lengths of the sides of a 30°-60°-90° triangle are in the ratio of 1 : √3: 2.

You can also recognize a 30°-60°-90° triangle by the angles. As long as you know that one of the angles in the right-angle triangle is either 30° or 60° then it must be a 30°-60°-90° special right triangle. A right triangle with a 30° angle or 60° angle must be a 30°-60°-90° special right triangle.

Side1 : Side2 : Hypotenuse = x : x√3 : 2x

Example 1:

Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are 4 inches and 3 inches.

Solution:

Step 1: Test the ratio of the lengths to see if it fits the n:n√2:2n ratio.

4:4√3:? = x:x√3:2x

Step 2: Yes, it is a 30°-60°-90° triangle for x = 4

Step 3: Calculate the third side.

2x = 2 × 4 = 8

Answer: The length of the hypotenuse is 8 inches.

What is a 3-4-5 Triangle?

A 3-4-5 triangle is a right triangle whose lengths are in the ratio of 3:4:5. When you are given the lengths of two sides of a right triangle, check the ratio of the lengths to see if it fits the 3:4:5 ratio.

Side1 : Side2 : Hypotenuse = 3n : 4n : 5n

Example 1:

Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are 6 inches and 8 inches.

Solution:

Step 1: Test the ratio of the lengths to see if it fits the 3n: 4n: 5n ratio.

6 : 8 : ? = 3(2) : 4(2) : ?

Step 2: It is a 3-4-5 triangle for n = 2.

Step 3: Calculate the third side.

5n = 5 × 2 = 10

Answer: The length of the hypotenuse is 10 inches.

Frequently Asked Questions

1. How do you solve right triangles?

Ans. To solve right triangles, you have to use the Pythagorean theorem. This states that:

a2 + b2 = c2

That means that if you know the lengths of two sides of a right triangle (a and b), you can find the length of the third side (c).

2. How do you solve a right triangle with one side?

Ans. To solve a right triangle with one side:

Choose the given side of the triangle as the hypotenuse, and label it “a”.

Let the other two sides of your triangle be labeled “b” and “c”, respectively.

Use the Pythagorean theorem to find the length of c: a2 + b2 = c2

The answer to your problem is C!

3. How do I find the angles of a right triangle?

Ans. The angles of a right triangle are found using the Pythagorean Theorem. This equation is a2 + b2 = c2, where a, b, and c are the lengths of the sides of the triangle.

In order to find an angle’s measure in a right triangle, you can use this formula:

sin(a) = opposite side / hypotenuse

cos(a) = adjacent side / hypotenuse

4. How do you find a right triangle with 3 sides?

Ans. To find a right triangle with three sides:

Draw a line from point A to point D.

Draw a line from point B to point C.

Draw a line from point C to point E.

The triangle formed by points A, B, and C is a right triangle if the sum of the lengths of its legs is equal to the length of its hypotenuse.

5. How to solve right triangle trigonometry?

Ans. Right triangle trigonometry can be tricky. Here’s how to do it:

1. Find the sine, cosine, or tangent of each angle in the triangle by using a calculator or plugging them into a chart.
2. Use these values to solve for unknown sides or angles of the triangle, using a calculator or plugging them into a chart.