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8. Right Triangles & Trigonometry
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Geometry

8.01 Right triangles and the Pythagorean theorem

Lesson
Practice

Right triangles and the Pythagorean theorem

A right angled triangle with hypotenuse and right angle labeled and opposite each other.

In a right triangle, the largest angle in the triangle is 90\degree. The side opposite the right-angle will be the longest side. We call this side the hypotenuse.

The other two shorter sides are called the legs of the triangle.

Exploration

A Pythagorean triple is a set of three positive integers that satisfy the Pythagorean theorem. A triangle, drawn with side lengths equivalent to a Pythagorean triple, will be a right triangle.

Here are two examples of Pythagorean triples.

Two right triangles with side lengths labeled and angles labeled theta. Ask your teacher for more information.

  1. Can you find another Pythagorean triple?

  2. How many do you think there are?

  3. Is there any easy way to find an infinite number of triples?

Pythagorean triple

Whole number side length measures of right triangles.

Example:

(3,\,4,\,5) and (5,\,12,\,13)

Given a Pythagorean triple, we can find another set by multiplying by a whole number to make a similar triangle with proportional sides.

Three right triangles: A B C, D E F and G H I. Ask your teacher for more information.

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

Pythagorean theorem

If \triangle ABC is a right triangle, then a^2+b^2=c^2.

Right triangle A B C with leg lengths a, b, and hypotenuse of c.
\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 a^2+b^2=c^2, then \triangle ABC is a right triangle.

If a triangle is not a right triangle, then the Pythagorean inequality theorem can be used to determine the type of triangle based on angles. Where c is the longest side, we have that:

  • If c^2<a^2+b^2, then the triangle is acute

  • If c^2>a^2+b^2, then the triangle is obtuse

Examples

Example 1

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
Two right triangles. The two triangles are connected with each other. Ask your teacher for more information.
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 solve for 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

Let's call common, unlabeled side b.

\displaystyle a^2+b^2\displaystyle =\displaystyle c^2Pythagorean theorem
\displaystyle 2^2+b^2\displaystyle =\displaystyle 5^2Substitution
\displaystyle 4+b^2\displaystyle =\displaystyle 25Evaluate the exponents
\displaystyle 21\displaystyle =\displaystyle b^2Combine like terms
\displaystyle \sqrt{21}\displaystyle =\displaystyle bEvaluate 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 4^2+\left(\sqrt{21}\right)^2\displaystyle =\displaystyle c^2Substitution
\displaystyle 16+21\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
Reflect and check

We can check the reasonableness of c by evaluating it on our calculator to get c=\sqrt{37}\approx 6.08.

Since c is the hypotenuse of the top triangle, it should be the longest side. We can also evaluate b=\sqrt{21}\approx 4.58, to see that c is indeed the longest side in the top triangle. We can see that c\approx 6.08 >4.58>4.

Similarly, the hypotenuse in the bottom triangle is the longest since 5>4.58>2.

Example 2

Select all of the Pythagorean triples.

A
\left(6,\,8,\,13\right)
B
\left(5,\,12,\,13\right)
C
\left(2,\,4,\,6\right)
D
\left(3,\,4,\,5\right)
Worked Solution
Create a strategy

Add the squares of the two smaller numbers. If the result is the same as the square of the biggest number, then it's a Pythagorean triple.

Apply the idea

The triple \left(a,\,b,\,c\right) is a Pythagorean triple if c^2=a^2+b^2.

\displaystyle 6^2 +8^2\displaystyle =\displaystyle 13^2Substitute the sides from option A
\displaystyle 100\displaystyle >\displaystyle 169Evaluate both sides
\displaystyle 5^2 +12^2\displaystyle =\displaystyle 13^2Substitute the sides from option B
\displaystyle 169\displaystyle =\displaystyle 169Evaluate both sides
\displaystyle 2^2 +4^2\displaystyle =\displaystyle 6^2Substitute the sides from option C
\displaystyle 20\displaystyle >\displaystyle 36Evaluate both sides
\displaystyle 3^2 +4^2\displaystyle =\displaystyle 5^2Substitute the sides from option D
\displaystyle 25\displaystyle =\displaystyle 25Evaluate both sides

The correct answers are options B and D.

Reflect and check

We may recognize \left(3,\,4,\,5\right) as a Pythagorean triple at this point. This means any of its scalar multiples will also be a Pythagorean triple, like \left(6,\,8,\,10\right). Based on this, we can eliminate \left(6,\,8,\,13\right) without doing calculations as if two sides are 6 and 8, the third must be 10, not 13.

Example 3

For the given triangle:

A triangle with side lengths of 9, 16, and 18.
a

Is the triangle a right triangle? Explain.

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

b

Classify the triangle in terms of its sides and angles.

Worked Solution
Create a strategy

From part (a), we know that a^2+b^2=337 and c^2=324, we can compare these to classify the angles in the triangle.

Apply the idea

We can see that all the sides are different lengths, so this is a scalene triangle.

Since c^2<a^2+b^2, using the Pythagorean inequality theorem, we can say that it is an acute triangle.

This triangle is an acute, scalene triangle.

Reflect and check

If the triangle were a right triangle we could also classify it this way. We could have a scalene right triangle or an isosceles right triangle, but we can never have an equilateral right triangle because the hypotenuse must always be longer than each leg.

Example 4

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. Ask your teacher for more information.

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

Start with the triangle with hypotenuse length of 304.6 \text{cm}, one leg with length 232.1 \text{cm}, and the other leg of length x \text{cm}.

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

x=197.26.

Next, look at the triangle with hypotenuse length of h \text{cm}, one leg with length x=197.26 \text{cm}, and the other leg of length 178.3 \text{cm}.

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

h=265.9 .

Reflect and check

When there are multiple triangles involved, we need to carefully determine if a length is a leg or a hypotenuse. A length may be a leg in one triangle and a hypotenuse in another.

Idea summary

Given two sides in a right triangle, we can find the third side length using the Pythagorean theorem.

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

Right triangle A B C with leg lengths a, b, and hypotenuse of c.

a^2+b^2=c^2Where a and b represent the lengths of the legs, and c represents the length of the hypotenuse.

The Pythagorean inequality theorem can be used to determine if a non-right triangle is acute or obtuse.

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

If a,\,b,\, and c satistfy a^2+b^2=c^2, and are positive integers, then we say that they form a Pythagorean triple. Any whole number scalar multiple of a Pythagorean triple is also a Pythagorean triple.

Outcomes

G.TR.4

The student will model and solve problems, including those in context, involving trigonometry in right triangles and applications of the Pythagorean Theorem.

G.TR.4a

Determine whether a triangle formed with three given lengths is a right triangle

G.TR.4g

Solve problems, including those in context, involving right triangles using the Pythagorean Theorem and its converse, including recognizing Pythagorean Triples.

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