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

8.01 Right triangles and the Pythagorean theorem

Lesson
Practice

Introduction

The Pythagorean theorem was introduced in 8th grade as a tool that we could use to find missing side lengths in right triangles. We will now use similar triangles to prove the theorem and use it to solve problems.

Right triangles and the Pythagorean theorem

Exploration

Consider the triangles shown in the diagram:

Right triangle uppercase A uppercase C uppercase B with right angle uppercase C. Point uppercase D is on uppercase A uppercase B. Segment uppercase C uppercase D is drawn perpendicular to uppercase A uppercase B. Uppercase A uppercase B has a length of lowercase c. Uppercase A uppercase C has a length of lowercase b. Uppercase C uppercase B has a length of lowercase a. Angle uppercase A uppercase B uppercase C is labeled theta.
  1. Which triangles in the diagram can we show are similar?
  2. How could we derive the Pythagorean theorem using ratios of corresponding sides of similar triangles?

The Pythagorean theorem and its converse describe how the side lengths of right triangles are related.

Pythagorean theorem

If a triangle is a right triangle, then the square of the length of the hypotenuse is equal to the sum of the squares of its leg lengths

For the given triangle, a^2+b^2=c^2

Right triangle uppercase A uppercase C uppercase B with right angle uppercase A uppercase C uppercase B. Uppercase A uppercase B has a length of lowercase c, uppercase A uppercase C has a length of lowercase b, and uppercase B uppercase C has a length of lowercase a.
\displaystyle a^2+b^2=c^2
\bm{a}
is the length of one of the legs (shorter sides) of the right triangle
\bm{b}
is the length of the other leg of the right triangle
\bm{c}
is the length of the hypotenuse of the right triangle
Converse of Pythagorean theorem

If the lengths a, b and c of the three sides of a triangle satisfy the relationship a^2+b^2=c^2, then the triangle is a right triangle

Pythagorean triple

A set of three non-zero whole numbers a, b, and c such that a^2+b^2=c^2

Example:

3, 4, and 5

Examples

Example 1

Given that \triangle{ABC} \sim \triangle {ACD} \sim \triangle{CBD}, find the length of \overline{BD} using properties of similar triangles.

Triangle A B C with right angle A C B. A point D is on A B. Segment C D is drawn and is perpendicular to A B. C D has a length of 16 and A D has a length of 32.
Worked Solution
Create a strategy

Use properties of similar triangles to write and solve a proportion and find the length of \overline{BD}. Since \triangle{ACD} \sim \triangle{CBD}, the corresponding side lengths are \overline{AD} and \overline{CD}, and \overline{CD} and \overline{BD}. We can write the proportion\dfrac{AD}{CD}=\dfrac{CD}{BD}

Apply the idea
\displaystyle \dfrac{AD}{CD}\displaystyle =\displaystyle \dfrac{CD}{BD}\triangle{ACD} \sim \triangle{CBD}
\displaystyle \dfrac{32}{16}\displaystyle =\displaystyle \dfrac{16}{BD}Substitution
\displaystyle 32BD\displaystyle =\displaystyle 256Multiply both sides of the equation by BD and 16
\displaystyle BD\displaystyle =\displaystyle 8Divide both sides of the equation by 32
Reflect and check

We can show that the corresponding side lengths of the similar triangles are proportional:\dfrac{32}{16}=\dfrac{16}{8}

Example 2

Consider the diagram shown below:

Triangle A B C with right angle A C B. A point D is on A B. Segment C D is drawn and is perpendicular to A B.
a

In \triangle{ABC}, altitude CD is drawn to its hypotenuse. Determine two triangles which must be similar to \triangle{ABC}.

Worked Solution
Create a strategy

Rotate the triangles so that they are oriented in the same direction and we can see corresponding side lengths:

Rotate \triangle{ACD} clockwise about A until \overline{C'A'} \parallel \overline{AD}.

Rotate \triangle{CBD} clockwise about B until \overline{C'B'} \parallel \overline{BD}.

Reflect \triangle{ABC} across a vertical line.

The three triangles from the diagram can be viewed together oriented in the same direction:

We can use AA similarity to determine two triangles that are similar to \triangle{ABC}.

Apply the idea

Since m \angle ACB = 90 \degree and m \angle CDB = 90 \degree, and since \angle ACB and \angle CDB are the same angle, m \angle ACB = m \angle CDB. \triangle{ABC} and \triangle{CBD} share \angle B.

Therefore, \triangle{CBD} \sim \triangle{ABC} by AA similarity.

Since m \angle ACB = 90 \degree and m \angle ADC = 90 \degree, and since \angle ACB and \angle ADC are the same angle, m \angle ACB = m \angle ADC. \triangle{ABC} and \triangle{ACD} share \angle A.

Therefore, \triangle{ACD} \sim \triangle{ABC} by AA similarity.

b

Using the similar triangles found in part (a), write ratios involving the corresponding parts (hypotenuses and legs) of the triangles.

Worked Solution
Create a strategy

Use the newly oriented triangles from part (a) to help write ratios comparing their side lengths.

Apply the idea

Since \triangle{ACD} \sim \triangle{ABC}, we have that

\displaystyle \dfrac{AB}{AC}\displaystyle =\displaystyle \dfrac{AC}{AD}Ratios of hypotenuses and shorter legs \triangle ABC and \triangle ACD

Since \triangle{CBD} \sim \triangle{ABC}, we have that

\displaystyle \dfrac{AB}{BC}\displaystyle =\displaystyle \dfrac{BC}{BD}Ratios of hypotenuses and longer legs \triangle ABC and \triangle CBD
\displaystyle \dfrac{AB}{BC}\displaystyle =\displaystyle \dfrac{BC}{AB-AD}Definition of collinear segments
c

Show that the ratios from part (b) lead to the Pythagorean theorem.

Worked Solution
Create a strategy

Recall the original triangle and use the ratios found in part (b) to prove that {AB}^2 = {AC}^2+ {BC}^2.

Apply the idea

Since \triangle{ACD} \sim \triangle{ABC}, we have that

\displaystyle \dfrac{AB}{AC}\displaystyle =\displaystyle \dfrac{AC}{AD}Ratios of hypotenuses and shorter legs \triangle ABC and \triangle ACD
\displaystyle {AB} \cdot {AD}\displaystyle =\displaystyle {AC}^2Multiply both sides by AC and AD

Since \triangle{CBD} \sim \triangle{ABC}, we have that

\displaystyle \dfrac{AB}{BC}\displaystyle =\displaystyle \dfrac{BC}{AB-AD}Ratios of hypotenuses and longer legs \triangle ABC and \triangle CBD
\displaystyle {AB}^2-AB \cdot BD\displaystyle =\displaystyle {BC}^2Multiply both sides by BC and AB-AD
\displaystyle {AB}^2-{BC}^2\displaystyle =\displaystyle AB \cdot ADSubtract {BC}^2 and add AB \cdot AD to both sides

Use that AB \cdot AD=AB \cdot AD:

\displaystyle {AB}^2-{BC}^2\displaystyle =\displaystyle {AC}^2Substitution
\displaystyle {AB}^2\displaystyle =\displaystyle {AC}^2+{BC}^2Add {BC}^2 to both sides

Example 3

Find the value of c for each triangle.

a
A right triangle with legs of length 14 and 48, and a hypotenuse of length c.
Worked Solution
Create a strategy

The triangle is a right triangle so the hypotenuse, c, can be found using the Pythagorean theorem.

Apply the idea
\displaystyle a^2+b^2\displaystyle =\displaystyle c^2Pythagorean theorem
\displaystyle 14^2+48^2\displaystyle =\displaystyle c^2Substitution
\displaystyle 196+2304\displaystyle =\displaystyle c^2Simplify the exponents
\displaystyle 2500\displaystyle =\displaystyle c^2Combine like terms
\displaystyle 50\displaystyle =\displaystyle cEvaluate the square root of both sides of the equation
Reflect and check

Notice that in this example, 2500 is a perfect square so our answer for c is an integer. This means \left\{ 14, 48, 50\right\} is a Pythagorean triple.

We can use the Pythagorean theorem to solve for missing leg lengths as well.

b
Worked Solution
Create a strategy

The value of c is the hypotenuse of the top triangle, and we need the leg lengths of the triangle to evaluate it. We can find the other missing leg length, which is one of the legs of the bottom triangle, using the Pythagorean theorem first and then use it to find the value of c.

Apply the idea
\displaystyle a^2+b^2\displaystyle =\displaystyle c^2Pythagorean theorem
\displaystyle 2^2+?^2\displaystyle =\displaystyle 5^2Substitution
\displaystyle 4+?\displaystyle =\displaystyle 25Evaluate the exponents
\displaystyle 21\displaystyle =\displaystyle ?^2Combine like terms
\displaystyle \sqrt{21}\displaystyle =\displaystyle ?Evaluate the square root of both sides of the equation

Since the length of the unknown leg of the top triangle is \sqrt{21}, we can use it with the other leg in the Pythagorean theorem to find the value of c as follows:

\displaystyle a^2+b^2\displaystyle =\displaystyle c^2Pythagorean theorem
\displaystyle \sqrt{21}^2+4^2\displaystyle =\displaystyle c^2Substitution
\displaystyle 21+16\displaystyle =\displaystyle c^2Evaluate the exponents
\displaystyle 37\displaystyle =\displaystyle c^2Combine like terms
\displaystyle \sqrt{37}\displaystyle =\displaystyle cEvaluate the square root of both sides of the equation

Example 4

Is the triangle below a right triangle? Explain.

A triangle with side lengths of 9, 16, and 18.
Worked Solution
Create a strategy

We want to use the converse of the Pythagorean theorem to determine if the triangle is a right triangle.

Let a and b represent the two shorter side lengths. The hypotenuse will be c.

Once we have labeled the sides, we want to find the value of a^2+b^2 and the value of c^2.

If a^2+b^2=c^2 then the triangle is a right triangle by the converse of the Pythagorean theorem.

If a^2+b^2 \neq c^2 then the triangle is not a right triangle for the same reason.

Apply the idea

Let a=9, b=16, and c=18. Now we can calculate the following:

\displaystyle a^2+b^2\displaystyle =\displaystyle 9^2+16^2
\displaystyle {}\displaystyle =\displaystyle 337

And

\displaystyle c^2\displaystyle =\displaystyle 18^2
\displaystyle {}\displaystyle =\displaystyle 324

This is not a right triangle because 337 \neq 324 so a^2+b^2 \neq c^2

Example 5

Archeologists have uncovered an ancient pillar which, after extensive digging, remains embedded in the ground. The lead researcher wants to record all of the dimensions of the pillar, including its height above the ground.

However, the team can only take certain measurements accurately without risking damage to the artifact. These measurements are shown in the diagram.

A diagram of a portion of a pillar in a diagonal position, and leaning to the right. A composite figure made up of a rectangle, and a right triangle is imposed on the portion of the pillar. The rectangle has a length of 178.3 centimeters, width of x centimeters, and a diagonal of h centimeters. The right triangle has side lengths of 232.1 centimeters, and x centimeters, and a hypotenuse of 304.6 centimeters.

Find the value of the variables.

Worked Solution
Create a strategy

The right triangle at the bottom of the pillar shows the length of a leg and its hypotenuse. Use the Pythagorean theorem to find the length of the unknown leg and its variable.

Then, the length of that unknown side is a leg of the triangle at the top of the pillar. Use the Pythagorean theorem to find the length of the unknown hypotenuse and its variable.

Apply the idea

Let the unknown side x=a.

\displaystyle a^2+b^2\displaystyle =\displaystyle c^2Pythagorean theorem
\displaystyle a^2+232.1^2\displaystyle =\displaystyle 304.6^2Substitution
\displaystyle a^2+53870.41\displaystyle =\displaystyle 92781.16Evaluate the exponents
\displaystyle a^2\displaystyle =\displaystyle 38910.75Subtract 53870.41 from both sides of the equation
\displaystyle a\displaystyle =\displaystyle 197.26Evaluate the square root of both sides of the equation

x=197.26.

Let the unknown hypotenuse for the top triangle h=c.

\displaystyle a^2+b^2\displaystyle =\displaystyle c^2Pythagorean theorem
\displaystyle 197.26^2+178.3^2\displaystyle =\displaystyle c^2Substitution
\displaystyle 70702.4\displaystyle =\displaystyle c^2Evaluate the exponents and addition
\displaystyle 265.9\displaystyle =\displaystyle cEvaluate the square root of both sides of the equation

h=265.9 .

Idea summary

Use the Pythagorean theorem and its converse to solve problems:

  • Pythagorean theorem: If a triangle is a right triangle, then the square of the length of the hypotenuse is equal to the sum of the squares of its leg lengths
  • Converse of the Pythagorean theorem: If the lengths a, b and c of the three sides of a triangle satisfy the relationship a^2+b^2=c^2, then the triangle is a right triangle

Outcomes

G.SRT.B.4

Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean theorem proved using triangle similarity.

G.SRT.C.8

Use trigonometric ratios and the Pythagorean theorem to solve right triangles in applied problems.

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