7. The Pythagorean Theorem and the Distance Formula
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Chapter 9
7.
The Pythagorean Theorem and the Distance Formula
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Why
The Pythagorean Theorem is a powerful concept in geometry used to relate the sides of right triangles. This lesson explores its applications, including how the theorem's converse helps determine if a triangle is a right triangle. It also introduces Pythagorean triples, which are sets of whole numbers satisfying the theorem, and explains the distance formula, which applies the theorem to find distances between points in a plane. These concepts are widely used in fields like engineering, navigation, and physics, where understanding spatial relationships and distances is essential.
Lesson Settings & Tools
| | 14 Theory slides |
| | 13 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
The Pythagorean Theorem and the Distance Formula
Slide of 14
Right triangles are found in everyday settings. Just look at the steps of stairs, a ladder leaning against a wall, or an inclined ramp. Additionally, right triangles are special in mathematics because their side lengths are related by a well-known equation. Continue on and learn about the awesome right triangle.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Challenge
Designing a Putting Green with Right Triangles
Paulina loves to play mini golf on her phone app. She plays against friends and she can even design her own golf course. The game has challenged her to design a square putting green surrounded by a sand bunker. The instructions are very specific.
Extra
How to Use the Applet- To move a triangle, click either the red vertex, the blue vertex, or inside the triangle and drag.
- To rotate a triangle, click on the yellow vertex and drag. The triangle will rotate about the blue vertex.
- Press the check button to verify whether the arrangement is correct.
- Press the hint button to get some help.
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Discussion
Relation Between the Side Lengths of a Right Triangle
Right triangles are special among all types of triangles. One reason for that is their side lengths meet a particular relationship, which is established in the Pythagorean Theorem.
Rule
Pythagorean Theorem
For right triangles, the length of the hypotenuse squared equals the sum of the squares of the lengths of the legs.
The theorem can be used to find the length of the third side when two side lengths are known.
Proof
Using AreaStart by drawing four congruent right triangles with legs a and b, and hypotenuse c. These triangles can be arranged to form two squares, with one square inside the other.
Notice that the side lengths of the outer square are equal to (a+b). Additionally, the side lengths of the inner square are equal to c. The area of both squares and the area of the four triangles are as follows.
| Area of the Inner Square | Area of the Outer Square | Area of the Four Triangles |
|---|---|---|
| c^2 | (a+b)^2 | 4* ab/2 = 2ab |
The area of the outer square equals the sum of the area of the inner square and the area of the four triangles. The previous diagram shows that. (a+b)^2=c^2+2ab This equation can be simplified by expanding the square of the binomial on the left-hand side.
(a+b)^2=c^2+2ab
PowToProdTwoFac
a^2=a* a
(a+b)(a+b)=c^2+2ab
Distr
Distribute (a+b)
(a+b)a+(a+b)b = c^2+2ab
Distr
Distribute a & b
a^2 + ab + ab + b^2 = c^2+2ab
AddTerms
Add terms
a^2+2ab+b^2=c^2+2ab
SubEqn
LHS-2ab=RHS-2ab
a^2+b^2=c^2 ✓
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Example
Mini Golf Tournament
Paulina is playing mini golf at a local course against a mighty rival — her mom, Momma Paulina. They are at the final hole 18 and Momma Paulina is wining by one stroke!
a Paulina is determined to beat Momma. Their friend acts as an announcer.
Paulina aims carefully. The ball is coming to a stop at the middle of the bridge! Wait. It is still going. It is rolling, rolling, and...it lands just 5 feet from the hole. What a stroke!
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b Their friend continues to act as an announcer.
Paulina's stroke has Momma sweating. She needs to focus if she wants to win this match. Here it goes. Not bad. Not bad at all. Momma's ball lands just 3 feet from Paulina's.
How far is Momma's ball from the hole?
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Hint
a Use the Pythagorean Theorem to find the length of the hypotenuse of the right triangle formed.
b Use the Pythagorean Theorem.
Solution
a Note that the tee box, the hole, and Paulina's ball form a right triangle. The right angle is formed at Paulina's ball.
The distance from the tee box to the hole is the length of the hypotenuse. That length can be found by using the Pythagorean Theorem. a^2 + b^2 = c^2 In this case, the legs are 5 and 25 feet long. Substitute these values into the equation. Then, solve it for c.
a^2 + b^2 = c^2
SubstituteII
a= 25, b= 5
25^2 + 5^2 = c^2
▼
Solve for c
CalcPow
Calculate power
625 + 25 = c^2
AddTerms
Add terms
650 = c^2
SqrtEqn
sqrt(LHS)=sqrt(RHS)
sqrt(650) = c
RearrangeEqn
Rearrange equation
c = sqrt(650)
c ≈ 25.5
The distance from the tee box to the hole is about 25.5 feet.
b Notice that the balls and the hole form a right triangle. The right angle is formed at Momma Paulina's ball. The balls are 3 feet apart.
The distance from Momma's ball to the hole is the length of a leg. However, the hypotenuse's length is also missing. Wait! From Part A, Paulina's ball is 5 feet from the hole. This means that the hypotenuse of the triangle is 5 feet long.
The missing length can be found by using the Pythagorean Theorem once more. a^2 + b^2 = c^2 Next, substitute 5 for c and 3 for either a or b. Then, solve the equation for the remaining variable.
Paulina's mom's ball is just 4 feet from the hole.
On the next stroke, Paulina put the ball in the hole. What a champ! Momma is currently winning by two strokes. If Momma puts it in on her very next stroke, she will win. If Momma misses, she needs to put the ball in on her next stroke to end in a tie. Otherwise, she will lose the game. What a thrilling match!
"Momma Paulina's final words say it all," exclaimed their announcer friend.
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Pop Quiz
Finding the Missing Length
Determine the missing side length of the given right triangle. Round the answer to two decimal places if needed.
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Discussion
Determining Whether a Triangle is a Right Triangle
The Pythagorean Theorem gives an equation that is only true if the triangle is a right triangle. However, swapping the hypothesis and the conclusion also makes a valid statement. This is known as the Converse of the Pythagorean Theorem.
Rule
Converse Pythagorean Theorem
Given a triangle, if the length of the longest side squared is equal to the sum of the squares of the other two side lengths, then the triangle is a right triangle. In this case, the right angle is opposite the longest side.
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Example
Flags
Paulina and her mom played an amazing game from start to finish. On their exit from the course, Paulina spots a very tall flag near the entrance that she did not notice earlier. Her mom says that it serves to indicate the wind direction to the golfers.
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Hint
Use the Converse of the Pythagorean Theorem.
Solution
A triangle is a right triangle if its side lengths meet the equation a^2+b^2=c^2 where c is the length of the longest side. This is thanks to the Converse of the Pythagorean Theorem.
The given triangle has 4, 5, and 6 meters long sides. Now, substitute these values into the equation to verify whether they satisfy it. Remember that c is the length of the longest side.
a^2+b^2=c^2
SubstituteValues
Substitute values
4^2 + 5^2 ? = 6^2
CalcPow
Calculate power
16 + 25 ? = 36
AddTerms
Add terms
41 ≠ 36
Since 41 is not equal to 36, the side lengths do not satisfy the equation. This implies that the triangle is not a right triangle.
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Pop Quiz
Is It a Right Triangle?
Determine whether the given triangle is a right triangle. Round the computations to one decimal place.
Extra
HintIt is enough to check if the length of the longest side squared is equal to the sum of the squares of the lengths of the other two sides.
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Discussion
Pythagorean Triple
A Pythagorean triple, commonly written as (a,b,c), is a set of three natural numbers that satisfy the Pythagorean Theorem. a^2+b^2=c^2 A right triangle can be drawn using the numbers of a Pythagorean triple as its side lengths. The lowest valued set of numbers that form a Pythagorean triple are 3, 4, and 5. There are infinitely many Pythagorean triples. The following table shows a few.
| Triple | Substitute in a^2+b^2=c^2 | Simplify |
|---|---|---|
| ( 3, 4, 5) | 3^2+ 4^2? = 5^2 | 9+16=25 ✓ |
| ( 5, 12, 13) | 5^2+ 12^2? = 13^2 | 25+144=169 ✓ |
| ( 8, 15, 17) | 8^2+ 15^2? = 17^2 | 64+225=289 ✓ |
| ( 7, 24, 25) | 7^2+ 24^2? = 25^2 | 49+576=625 ✓ |
If (a,b,c) is a Pythagorean triple, then so is (ka,kb,kc) for any natural number k.
| (a,b,c) | k | (ka,kb,kc) |
|---|---|---|
| ( 3, 4, 5) | 3 | ( 3( 3), 3( 4), 3( 5)) ⇕ (9,12,15) |
| ( 5, 12, 13) | 2 | ( 2( 5), 2( 12), 2( 13)) ⇕ (10,24,26) |
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Pop Quiz
Studying Triples
Consider the numbers in the following table and answer what is required. Keep in mind that the numbers are always ordered from smallest to greatest.
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Discussion
Distance Between Two Points
In many situations it is useful to know the distance between two objects. If those objects are plotted on a coordinate plane, the Distance Formula can be used to find their distance.
Rule
Distance Formula
Given two points A(x_1, y_1) and B(x_2, y_2) on a coordinate plane, their distance d is given by the following formula.
d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
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Example
A Long Stroke
On the drive home, Paulina breaks out a game of golf on her smartphone. She is about to take her first stroke using the driver golf club. This club is used for hitting the ball the farthest down the fairway. There she goes!
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b The putter is a golf club that is used when the ball is on the putting green, in this case, less than 15 yards from the hole. The hole is located at (70,51). Should Paulina use the putter for her next stroke?
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Hint
a Use the distance formula.
b Find the distance between the ball and the hole. If the distance is less than 15 yards, the ball is on the putting green.
Solution
a According to the app, the tee box is located at T(-10,5). The ball, after Paulina's stroke, is at B(60,40). Graph these points on a coordinate plane.
The distance traveled by the ball is the distance between these two points. It can be found by using the distance formula. d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2) Substitute the coordinates of points T and B into the formula and simplify.
d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
SubstitutePoints
Substitute ( -10,5) & ( 60,40)
d = sqrt(( 60-( -10))^2 + ( 40- 5)^2)
SubNeg
a-(- b)=a+b
d = sqrt((60+10)^2 + (40-5)^2)
AddSubTerms
Add and subtract terms
d = sqrt(70^2 + 35^2)
CalcPow
Calculate power
d = sqrt(4900 + 1225)
AddTerms
Add terms
d = sqrt(6125)
UseCalc
Use a calculator
d = 78.262 37...
RoundDec
Round to 1 decimal place(s)
d ≈ 78.3
The ball traveled about 78.3 yards.
b If a ball is less than 15 yards away from the hole, it is on the putting green. Calculate the distance between the ball and the hole. Begin by drawing the given points on a coordinate plane.
The distance is found by using the distance formula.
d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
SubstitutePoints
Substitute ( 60,40) & ( 70,51)
d = sqrt(( 70- 60)^2 + ( 51- 40)^2)
SubTerms
Subtract terms
d = sqrt(10^2 + 11^2)
CalcPow
Calculate power
d = sqrt(100 + 121)
AddTerms
Add terms
d = sqrt(221)
UseCalc
Use a calculator
d = 14.866 06...
The ball is about 14.9 yards away from the hole. Since it is less than 15 yards, the ball is on the putting green. Paulina should use the putter for her next stroke.
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Pop Quiz
Finding the Distance Between Two Points
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Closure
Contributions by Pythagoras
The Pythagorean Theorem is named after the Greek philosopher and mathematician Pythagoras, who lived in the 6th century BC. He is known as the founder of the Pythagorean school and is considered one of the most influential thinkers of the ancient world.
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The Pythagorean Theorem and the Distance Formula
Exercise 2.1
Tadeo's age together with the ages of his parents form a Pythagorean triple. Tadeo is 12 years old, his mother is 35 years old, and his father is older than his mother. How old is Tadeo's father?
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Let's begin by recalling what a Pythagorean triple is.
Pyhtgaorean Triple |- A set of three natural numbers that satisfy the Pythagorean Theorem.
We know that Tadeo is 12 years old and his mother is 35 years old. Let x be Tadeo's father age. We are said that their ages form a Pythagorean triple. This means that these three numbers satisfy the following equation. a^2+b^2=c^2 In this formula, c is the greatest of the three numbers. Since Tadeo's father is older than Tadeo's mother, we will substitute x for c. Also, we will substitute the other two ages for a and b.
Tadeo's father is 37 years old.
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Solution
The shortest leg of a right triangle is 65 millimeters long and its hypotenuse is 97 millimeters long. What is the perimeter of the triangle?
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The perimeter of a polygon is the sum of all its side lengths. In our case, we are given a right triangle and two side lengths. We need to find the third side length to calculate its perimeter. Let's begin by making a sketch of the described right triangle.
We can find the length of AB by applying the Pythagorean Theorem. Let's substitute 97 for c and 65 for a.
The length of AB is 72 millimeters. Now that we know the three side lengths, we can calculate the perimeter of the triangle.
The perimeter of the described right triangle is 234 millimeters.
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Solution
Do the following points form a right triangle? A(-3,-2), B(-4,3), C(3,-1)
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We can begin by graphing the three points on a coordinate plane to visually estimate whether they form a right triangle.
At first glance, ∠ A seems to be a right angle. If that is true, then △ ABC is a right triangle. However, we cannot ensure it by the graph. One way to determine if the triangle is a right triangle is through the Converse of the Pythagorean Theorem.
Converse of the Pythagorean Thereom |- Given a triangle, if the length of the longest side squared is equal to the sum of the squares of the other two side lengths, then the triangle is a right triangle.
Let's find the side lengths of △ ABC by using the distance formula. d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2) We can begin by finding AC. Let's substitute the coordinates of the points A(-3,-2) and C(3,-1) into the formula and keep the results in radical form.
We can find the other two side lengths following a similar procedure. The computations are summarized in the following table.
| Points | d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2) | Simplify | Length |
|---|---|---|---|
| A( -3, -2) and B( -4, 3) | AB = sqrt(( -4-( -3))^2+( 3-( -2))^2) | AB=sqrt((-1)^2+5^2) | AB = sqrt(26) |
| B( -4, 3) and C( 3, -1) | BC = sqrt(( 3-( -4))^2+( -1- 3)^2) | BC=sqrt(7^2+(-4)^2) | BC = sqrt(65) |
Now that we know the three side lengths, let's substitute them into the equation given by the Pythagorean Theorem. If we get a true statement, the points form a right triangle; otherwise, they do not. Remember, c corresponds to the greatest length.
We got a false statement, 63 is not equal to 65. Therefore, the points do not form a right triangle. We can use a protractor to determine the measure of ∠ A.
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Solution
Find the perimeter of the following triangle.
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The perimeter of a polygon is the sum of all its side lengths. In our case, we are not given any side length. However, we can use the coordinates of the vertices and the distance formula to calculate each side length. Let's begin by identifying the vertices' coordinates.
Now, let's recall the distance formula. d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2) We can start by finding the distance between A and B.
The other two side lengths are determined in a similar fashion. The computations are summarized in the following table.
| Points | d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2) | Simplify | Side Length |
|---|---|---|---|
| B( 2, 3) and C( -4, 3) | BC = sqrt(( -4- 2)^2 + ( 3- 3)^2) | BC=sqrt((-6)^2 +0^2) | BC = 6 |
| A( -1, -1) and C( -4, 3) | AC = sqrt(( -4-( -1))^2 + ( 3-( -1))^2) | AC=sqrt((-3)^2 +4^2) | AC = 5 |
Now that we know the three side lengths, it is time to find the perimeter of △ ABC by adding them!
The perimeter of the triangle is 16.
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Solution
The Pythagorean Theorem and the Distance Formula
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