Topics
8. Triangle Trigonometry
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United States of America
Grades 9-12

8.02 Special right triangles

Lesson
Practice

Introduction

This lesson gives us an opportunity to apply our reasoning about similar figures and our tools from lesson  8.01 Right triangles and the Pythagorean theorem  to make sense of a relationship in two special triangles. These two types of triangles form special sets of similar triangles with a constant ratio of their respective side lengths.

Special right triangles

Exploration

Drag the points to change the segments on the applet.

Loading interactive...
  1. Drag one of the sliders. How would you classify the triangle that forms?
  2. What are the angle measures of the triangle?
  3. If the length of the base is 1 unit, what are the lengths of the hypotenuse and the other leg?
  4. Repeat questions 1-3 with the other triangle.

While the Pythagorean theorem can apply to any kind of right triangle, there are particular types of right triangles whose side lengths and angles have helpful properties.

45 \degree- 45 \degree- 90\degree triangle theorem

In a 45 \degree- 45 \degree- 90\degree triangle, the legs are congruent and the length of the hypotenuse is \sqrt{2} times the length of the legs.

Ratio of sides \quad 1:1: \sqrt{2}

Right triangle A B C with right angle A. The legs of the triangle have a length of s, and the hypotenuse has a length of s square root of 2. Angles B and C have a measure of 45 degrees.
30 \degree- 60 \degree- 90\degree triangle theorem

In a 30 \degree- 60 \degree- 90\degree triangle, the length of the hypotenuse is twice the length of the short leg. The length of the long leg is \sqrt{3} times the length of the short leg.

Ratio of sides \quad 1:\sqrt{3}: 2

Right triangle A B C with right angle C. Leg B C has length of s, leg A C has length of s square root of 3, and hypotenuse A B has a length of 2 s. Angle A has a measure of 30 degrees, and angle B has a measure of 60 degrees.

Examples

Example 1

Consider the triangle below:

Right triangle with both legs of length 1, and two angles measuring 45 degrees.
a

Find the length of the hypotenuse.

Worked Solution
Create a strategy

Call the hypotenuse c. Use the Pythagorean theorem to find the missing side of the triangle.

Apply the idea
\displaystyle a^2+b^2\displaystyle =\displaystyle c^2Pythagorean theorem
\displaystyle 1^2+1^2\displaystyle =\displaystyle c^2Substitution
\displaystyle 2\displaystyle =\displaystyle c^2Evaluate the exponents and addition
\displaystyle \sqrt{2}\displaystyle =\displaystyle cEvaluate the square root of both sides of the equation
b

Find and justify the ratio of proportionality between the side lengths of any 45 \degree- 45 \degree- 90\degree triangle.

Worked Solution
Create a strategy

Draw a triangle with the given angle measures.

Right triangle with two angles measuring 45 degrees.
Apply the idea

All 45 \degree- 45 \degree- 90\degree triangles are similar because they have three corresponding congruent angles. We also know a 45 \degree- 45 \degree- 90\degree triangle is an isosceles triangle by the converse of the base angles theorem. Therefore, its legs are congruent.

Let the legs of the triangle be x. Call the hypotenuse of the triangle y.

Right triangle with both legs of length x, hypotenuse of length y, and two angle measuring 45 degrees.

We can use the Pythagorean theorem to show that:

\displaystyle a^2+b^2\displaystyle =\displaystyle c^2Pythagorean theorem
\displaystyle x^2+x^2\displaystyle =\displaystyle y^2Substitution
\displaystyle 2x^2\displaystyle =\displaystyle y^2Combine like terms
\displaystyle x \sqrt{2}\displaystyle =\displaystyle yEvaluate the square root of both sides of the equation

Using this, we can say that the ratio of the side lengths of any 45 \degree- 45 \degree- 90\degree triangle is x : x : x \sqrt{2}, or 1:1: \sqrt{2}.

Example 2

Consider the triangle below:

Right triangle with both legs of length 2, and both base angles measuring 60 degrees. A vertical segment is drawn from the apex to the base of the triangle. This segment divides the remaining angle of the triangle into two, both measuting 30 degrees.
a

Find the height of the triangle.

Worked Solution
Create a strategy

Equiangular triangles are also equilateral, so the base is also 2 units. The altitude of an equilateral triangle is also the median, so half of the base must be 1 unit. We can use the Pythagorean Theorem to find the missing side of the triangle, which we will call b.

Right triangle with one leg of length 1, hypotenuse length of 2. The angle opposite the side of length 1 has a measure of 30 degrees. The adjacent angle has a measure of 60 degrees.
Apply the idea
\displaystyle a^2+b^2\displaystyle =\displaystyle c^2Pythagorean theorem
\displaystyle 1^2+b^2\displaystyle =\displaystyle 2^2Substitution
\displaystyle 1 + b^2\displaystyle =\displaystyle 4Evaluate the exponents
\displaystyle b^2\displaystyle =\displaystyle 3Subtract 1 from both sides of the equation
\displaystyle b\displaystyle =\displaystyle \sqrt{3}Evaluate the square root of both sides of the equation
b

Find and justify the ratio of proportionality between the side lengths of any 30 \degree- 60 \degree- 90\degree triangle.

Worked Solution
Create a strategy

Draw and label an equilateral triangle with a 30 \degree- 60 \degree- 90\degree triangle drawn.

An equilateral triangle with an angle bisector drawn. The angle bisector is also the height of the triangle.
Apply the idea

All 30 \degree- 60 \degree- 90\degree triangles are similar because they have three corresponding congruent angles. We also know a 30 \degree- 60 \degree- 90\degree triangle is half of an equilateral triangle.

Let the short leg of the triangle be x. Then the hypotenuse of the triangle must be 2x. Call the missing leg of the triangle y.

We can use the Pythagorean theorem to show that:

\displaystyle a^2+b^2\displaystyle =\displaystyle c^2Pythagorean theorem
\displaystyle x^2+y^2\displaystyle =\displaystyle 4x^2Substitution and evaluate the exponent
\displaystyle y^2\displaystyle =\displaystyle 3x^2Subtract x^2 from both sides of the equation
\displaystyle y\displaystyle =\displaystyle x \sqrt{3}Evaluate the square root of both sides of the equation and apply the commutative property of multiplication

Using this, we can say that the ratio of the side lengths of any 30 \degree- 60 \degree- 90\degree triangle is x : x \sqrt{3} : 2x, or 1:\sqrt{3}:2.

Example 3

Consider the triangle below:

A right triangle with legs of length a and 15, and a hypotenuse of length c. The angle opposite the leg with length a has a measure of 45 degrees.
a

Find the exact value of a.

Worked Solution
Create a strategy

The triangle has a right angle and a 45\degree angle shown. This means that this triangle is a 45 \degree- 45 \degree- 90\degree triangle because the sum of interior angles in a triangle is equal to 180 \degree.

Apply the idea

Legs in 45 \degree- 45 \degree- 90\degree special right triangles are congruent, and the ratio of the sides is 1:1: \sqrt{2}. We are given that 15 is the length of one leg from the diagram.

a is the other leg of the triangle as a is opposite a 45 \degree angle.

So, a=15.

b

Find the exact value of c.

Worked Solution
Create a strategy

Using the 45 \degree- 45 \degree- 90\degree triangle theorem, the hypotenuse is \sqrt{2} times the length of the leg. We want to use this property to find c.

Apply the idea

We know the length of the legs is 15 and c is the hypotenuse as it is opposite the right angle.

So, c=15\sqrt{2}.

Reflect and check

We could have also used the Pythagorean theorem to solve for c.

\displaystyle a^2 + b^2\displaystyle =\displaystyle c^2Pythagorean theorem
\displaystyle 15^2+15^2\displaystyle =\displaystyle c^2Substitute the side lengths a=15 and b=15
\displaystyle 450\displaystyle =\displaystyle c^2Evaluate the exponents and addition
\displaystyle 21.21\displaystyle =\displaystyle cEvaluate the square root of both sides of the equation

15\sqrt{2} \approx 21.21.

Example 4

Find the value of each variable in the diagram shown. Round your answer to two decimal places.

Worked Solution
Create a strategy

Both triangles are special 30 \degree- 60 \degree- 90\degree triangles. Use that the side lengths of a 30 \degree- 60 \degree- 90\degree triangle have a ratio of 1:\sqrt{3}: 2 and the Pythagorean theorem to find each missing side length of each triangle.

Apply the idea

The hypotenuse of the bottom triangle is 6 units, so the ratio 1:\sqrt{3}: 2 would lead the ratio of the hypotenuse to the short leg of the triangle to be 2:1. This means that the short leg of the bottom triangle, r=3.

Using the Pythagorean theorem for the bottom triangle, we have:

\displaystyle a^2+b^2\displaystyle =\displaystyle c^2Pythagorean theorem
\displaystyle 3^2+s^2\displaystyle =\displaystyle 6^2Substitution
\displaystyle 9+s^2\displaystyle =\displaystyle 36Evaluate the exponents
\displaystyle s^2\displaystyle =\displaystyle 27Subtract 9 from both sides of the equation
\displaystyle s\displaystyle =\displaystyle 5.2Evaluate the square root of both sides of the equation

The longest leg of the top triangle is given as 6 units, so the ratio 1:\sqrt{3}: 2 means that the longest leg is \sqrt{3} times the length of the short leg, and the hypotenuse will be twice the length of the short leg. So we have:

\displaystyle p\sqrt{3}\displaystyle =\displaystyle 6The length of the longest leg is \sqrt{3} times the length of the short leg
\displaystyle p\displaystyle =\displaystyle 3.46Divide both sides of the equation by \sqrt{3}

Finally, since we know that the hypotenuse is twice the length of the short leg, we have:

\displaystyle q\displaystyle =\displaystyle 2p
\displaystyle q\displaystyle =\displaystyle 2 \left( 3.46 \right)Substitution
\displaystyle q\displaystyle =\displaystyle 6.92Evaluate the multiplication
Reflect and check

For the first triangle, we could have found s with the ratio, since we knew that r=3. Since s, or the longest leg, is \sqrt{3} times the length of the short leg, r, we know that s= 3 \sqrt{3}. This is approximately 5.2, which we calculated using the Pythagorean theorem above.

Idea summary

We can use special right triangles to find missing side lengths in right triangles:

Right triangle A B C with right angle A. The legs of the triangle have a length of s, and the hypotenuse has a length of s square root of 2. Angles B and C have a measure of 45 degrees.

In a 45 \degree- 45 \degree- 90\degree triangle, the legs are congruent and the length of the hypotenuse is \sqrt{2} times the length of the legs.

Ratio of sides \quad 1:1: \sqrt{2}

Right triangle A B C with right angle C. Leg B C has length of s, leg A C has length of s square root of 3, and hypotenuse A B has a length of 2 s. Angle A has a measure of 30 degrees, and angle B has a measure of 60 degrees.

In a 30 \degree- 60 \degree- 90\degree triangle, the length of the hypotenuse is twice the length of the short leg. The length of the long leg is \sqrt{3} times the length of the short leg.

Ratio of sides \quad 1:\sqrt{3}: 2

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