What do you notice about the sum of the area of the squares of the two shorter sides and the area of the square of the hypotenuse?
If you subtracted the areas of the shorter sides' squares from the area of the largest side's square, what do you get?
Can you find another set of sides, where all three sides have integer values?
Try to write an equation that will always be true using the variables a,\,b,\, and c.

The Pythagorean theorem says that in a right triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides:

\displaystyle a^2+b^2=c^2

\bm{c}

is the length of the hypotenuse

\bm{a}

is one of the shorter side lengths

\bm{b}

is the other shorter side length

A right triangle having side lengths of a, b and c where c is the hypotenuse.

We can use this formula to find the length of any side of a right triangle if we know the lengths of the other two.

The converse of the Pythagorean Theorem says that if the square of the length of the hypotenuse equals the sum of the squares of the legs in a triangle, then the triangle is a right triangle. This can be used to determine whether a triangle is a right triangle given the measures of its three sides.

A right triangle has side lengths 13, 12, and 5. Ask your teacher for more information.

Let's determine if this is a right triangle.

We know the hypotenuse, c, is the longest side, and our legs, a,\,b, are the shorter sides.

a^2+b^2=c^2

5^2 + 12^2=13^2

25+144=169

169=169

Yes, this is a right triangle.

If a group of 3 integers can create the sides of a right triangle, we call this group of numbers a Pythagorean triple. This group of numbers (a,\,b,\,c) will satisfy the Pythagorean theorem: a^2+b^2=c^2.

(6,\,8,\,10) is a Pythagorean triple because:

If we know one Pythagorean triple we can create others using multiplication or division. 6, \,8 and 10 have a common factor of 2. If we divide each number in the triple by this common factor, we get another Pythagorean triple (3,\,4,\,5).

Multiplying (3,\,4,\,5) by 5 gives us another Pythagorean triple (15,\,20,\,25).

Two right angles. The small triangle has side lengths of 5, 4, and 3. The arrow pointing the big triangle labeled x 5. The big triangle has side lengths of 25, 20, and 15.

What we've really done here is use a scale factor to create a similar triangle by applying a dilation. Remember, similar triangles have corresponding congruent angles and proportional corresponding sides. So if we create a new triangle that is similar to a right triangle, the new triangle must have a right angle and so it must be a right triangle.

Examples

Example 1

Determine if the triangle below is a right triangle. If so, label the hypotenuse and legs.

Triangle XYZ have side lengths 12 meters, 16 meters, and 20 meters. Ask your teacher for more information.

Worked Solution

Create a strategy

Using the longest side as the hypotenuse, substitute the values into the Pythagorean Theorem to see if it results in a true statement.

Apply the idea

From the given triangle we will use a=12,\,b=16,\, and c=20.

\displaystyle c^2	\displaystyle =	\displaystyle a^2+b^2	Use the Pythagorean theorem
\displaystyle 20^2	\displaystyle =	\displaystyle 12^2+16^2	Substitute the values
\displaystyle 400	\displaystyle =	\displaystyle 144+256	Evaluate the exponents
\displaystyle 400	\displaystyle =	\displaystyle 400	Evaluate the addition

Yes, this is a right triangle.

Triangle XYZ have side lengths 12 meters, 16 meters, and 20 meters. The legs and the hypotenuse are labeled.

Example 2

Find the length of the hypotenuse, c in this triangle.

A right triangle having side length of 15, 8 and unknown side of c. Ask your teacher for more information.

Worked Solution

Create a strategy

Use the Pythagorean theorem: c^2=a^2+b^2.

Apply the idea

Based on the given diagram, we are given a=8 and b=15.

\displaystyle c^2	\displaystyle =	\displaystyle a^2+b^2	Use the Pythagorean theorem
	\displaystyle =	\displaystyle 8^2+15^2	Substitute the values
	\displaystyle =	\displaystyle 64+ 225	Evaluate the squares
	\displaystyle =	\displaystyle 289	Evaluate
\displaystyle \sqrt{c^2}	\displaystyle =	\displaystyle \sqrt{289}	Apply the square root to both sides
\displaystyle c	\displaystyle =	\displaystyle 17\, \text{cm}	Evaluate the square root

Reflect and check

Because all the lengths for the sides of this triangle are whole numbers and they satisfy Pythagorean theorem, the numbers (8,\,15,\,17) form a Pythagorean triple.

Example 3

Calculate the value of a in the triangle below.

A right triangle having side lengths 15 centimeters and an unknown side of a. The hypotenuse is of 17 centimeters.

Worked Solution

Create a strategy

Use the Pythagorean theorem: c^2=a^2+b^2\,

Apply the idea

We are given b=15 and c=17.

\displaystyle c^2	\displaystyle =	\displaystyle a^2+b^2	Use the Pythagorean theorem
\displaystyle 17^2	\displaystyle =	\displaystyle a^2+15^2	Substitute the values
\displaystyle 17^2 -15^2	\displaystyle =	\displaystyle a^2	Subtract 15^2 from both sides of the equation
\displaystyle 289 - 225	\displaystyle =	\displaystyle a^2	Evaluate the squares
\displaystyle 64	\displaystyle =	\displaystyle a^2	Evaluate the subtraction
\displaystyle \sqrt{64}	\displaystyle =	\displaystyle \sqrt{a^2}	Apply the square root to both sides
\displaystyle 8	\displaystyle =	\displaystyle a	Evaluate the square root

Example 4

A triangle has side lengths of 6,\,7,\, and 11. Do these three side lengths make a Pythagorean triple?

Worked Solution

Create a strategy

Using the longest side as the hypotenuse and substitute the values into the Pythagorean Theorem to see if we get a true statement.

Apply the idea

We will use a=6,\,b=7,\, and c=11.

\displaystyle c^2	\displaystyle =	\displaystyle a^2+b^2	Use the Pythagorean theorem
\displaystyle 11^2	\displaystyle =	\displaystyle 6^2+7^2	Substitute the values
\displaystyle 121	\displaystyle =	\displaystyle 36+49	Evaluate the exponents
\displaystyle 121	\displaystyle \neq	\displaystyle 85	Evaluate the addition

No, this is not a right triangle so the numbers cannot be a Pythagorean triple.

Example 5

Luke knows the two largest numbers in a Pythagorean Triple, which are 37 and 35. What number, a, does Luke need to complete the triple?

Worked Solution

Create a strategy

Use the Pythagorean theorem : c^2 = a^2+b^2.

Apply the idea

We are given b=35 and c=37.

\displaystyle c^2	\displaystyle =	\displaystyle a^2+b^2	Use the Pythagorean theorem
\displaystyle 37^2	\displaystyle =	\displaystyle a^2+35^2	Substitute the values
\displaystyle 37^2-35^2	\displaystyle =	\displaystyle a^2	Subtract 35^2 from both sides of the equation
\displaystyle 1369 - 1225	\displaystyle =	\displaystyle a^2	Evaluate the squares
\displaystyle 144	\displaystyle =	\displaystyle a^2	Evaluate the subtraction
\displaystyle \sqrt{144}	\displaystyle =	\displaystyle {a^2}	Apply the square root to both sides
\displaystyle 12	\displaystyle =	\displaystyle a	Evaluate the square root

So, the number that will complete the triple is 12.

Example 6

A sports association wants to redesign the trophy they award to the player of the season. The front view of one particular design is shown in the diagram:

A trophy made from 2 right angled triangles joined at the hypotenuse. Ask your teacher for more information.

Find the value of x.

Worked Solution

Create a strategy

We can use the Pythagorean Theorem: a^{2}+b^{2}=c^{2}.

Apply the idea

We can apply Pythagorean theorem to the bottom triangle that has side lengths of 16\text{ cm} and 12\text{ cm}, since the only unknown side is the hypotenuse.

\displaystyle 12^{2}+16^{2}	\displaystyle =	\displaystyle x^{2}	Substitute a=12,\,b=16,\,and c=x
\displaystyle 144+256	\displaystyle =	\displaystyle x^{2}	Evaluate the squares
\displaystyle 400	\displaystyle =	\displaystyle x^{2}	Evaluate the addition
\displaystyle \sqrt{400}	\displaystyle =	\displaystyle x	Take the square root of both sides
\displaystyle 20\text{ cm}	\displaystyle =	\displaystyle x	Evaluate the square root

Find the value of y, rounded to two decimal places.

Worked Solution

Create a strategy

We can use the Pythagorean Theorem: a^{2} + b^{2}=c^{2}.

Apply the idea

\displaystyle y^{2} + 3 ^{2}	\displaystyle =	\displaystyle 20^{2}	Substitute a=y,\,b=3,\,and c=20
\displaystyle y^{2}+9	\displaystyle =	\displaystyle 400	Evaluate the squares
\displaystyle y^{2} + 9 -9	\displaystyle =	\displaystyle 400-9	Subtract 9 from both sides
\displaystyle y^{2}	\displaystyle =	\displaystyle 391	Evaluate the subtraction
\displaystyle y	\displaystyle =	\displaystyle \sqrt{391}	Take the square root of both sides
\displaystyle y	\displaystyle =	\displaystyle 19.77\text{ cm}	Evaluate and round to two decimal places

Idea summary

The Pythagorean theorem states that in a right triangle the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides:

\displaystyle a^2+b^2=c^2

\bm{c}

is the length of the hypotenuse

\bm{a}

is one of the shorter side lengths

\bm{b}

is the other shorter side length

A Pythagorean triple is an ordered triple (a,\,b,\,c) of three positive integers that represent the side lengths of a right triangle.

Outcomes

8.MG.4

The student will apply the Pythagorean Theorem to solve problems involving right triangles, including those in context.

8.MG.4a

Verify the Pythagorean Theorem using diagrams, concrete materials, and measurement.

8.MG.4b

Determine whether a triangle is a right triangle given the measures of its three sides.

8.MG.4c

Identify the parts of a right triangle (the hypotenuse and the legs) given figures in various orientations.

8.MG.4d

Determine the measure of a side of a right triangle, given the measures of the other two sides.

8.MG.4e

Apply the Pythagorean Theorem, and its converse, to solve problems involving right triangles in context.

6.02 Pythagorean theorem

Ideas

Pythagorean Theorem

Exploration

Examples

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Outcomes

8.MG.4

8.MG.4a

8.MG.4b

8.MG.4c

8.MG.4d

8.MG.4e

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