Understanding the Pythagorean Theorem and the Distance Formula

Area of the Inner Square	Area of the Outer Square	Area of the Four Triangles
$c^{2}$	$(a + b)^{2}$	$4 \cdot \frac{a b}{2} = 2 a b$

Area of the Inner Square

Area of the Outer Square

Area of the Four Triangles

c^{2}

(a + b)^{2}

4 \cdot \frac{a b}{2} = 2 a b

Triple	Substitute in $a^{2} + b^{2} = c^{2}$	Simplify
$(3, 4, 5)$	$3^{2} + 4^{2} = ? 5^{2}$	$9 + 16 = 25 ✓$
$(5, 12, 13)$	$5^{2} + 12^{2} = ? 13^{2}$	$25 + 144 = 169 ✓$
$(8, 15, 17)$	$8^{2} + 15^{2} = ? 17^{2}$	$64 + 225 = 289 ✓$
$(7, 24, 25)$	$7^{2} + 24^{2} = ? 25^{2}$	$49 + 576 = 625 ✓$

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 ✓

$(a, b, c)$	$k$	$(k a, k b, k c)$
$(3, 4, 5)$	$3$	$(3 (3), 3 (4), 3 (5))$ $⇕$ $(9, 12, 15)$
$(5, 12, 13)$	$2$	$(2 (5), 2 (12), 2 (13))$ $⇕$ $(10, 24, 26)$

(a, b, c)

k

(k a, k b, k c)

(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)

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?

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.

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?

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.

Do the following points form a right triangle?

A (- 3, - 2), B (- 4, 3), C (3, - 1)

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.

Find the perimeter of the following triangle.

Triangle with vertices (-1,-1), (2,3), and (-4,3).

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.

The Pythagorean Theorem and the Distance Formula

Catch-Up and Review

Designing a Putting Green with Right Triangles

Extra

Relation Between the Side Lengths of a Right Triangle

Pythagorean Theorem

Proof

Mini Golf Tournament

Hint

Solution

Finding the Missing Length

Determining Whether a Triangle is a Right Triangle

Converse Pythagorean Theorem

Flags

Hint

Solution

Is It a Right Triangle?

Extra

Pythagorean Triple

Studying Triples

Distance Between Two Points

Distance Formula

A Long Stroke

Hint

Solution

Finding the Distance Between Two Points

Contributions by Pythagoras

The Pythagorean Theorem and the Distance Formula

Recommended exercises

	14 Theory slides
	13 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions