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)

Find the missing side length of each of the following triangles.

A right triangle with legs 6 and 8 units long.

A right triangle with hypotenuse 13 units long and one leg 5 units long.

We begin by noticing that △ ABC is a right triangle. We are given the length of its legs, 6 and 8 units long, and we need to find the length of the hypotenuse. We can do this by applying the Pythagorean Theorem. a^2+b^2=c^2 In this formula, a and b are the lengths of the legs and c is the length of the hypotenuse of a right triangle. Let's substitute 6 and 8 for a and b. Then, we will solve the equation for c.

The hypotenuse of the triangle is 10 units long. This means that BC=10.

We have that △ DEF is a right triangle of which we know the length of the hypotenuse and the length of one leg. We need to find the length of the other leg. We can use the Pythagorean Theorem again. a^2+b^2=c^2 Let's substitute 5 and 13 for a and c, respectively. Then, we will solve the equation for b.

The missing length is 12. This means that DF = 12.

Determine whether the given triangle is a right triangle.

Triangle

P Q R

has sides of lengths

13,

3,

and

2 .

We can determine whether a triangle is a right triangle using the Converse of the Pythagorean Theorem. Let's start by recalling what this theorem says.

Converse of the 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 our case, the given triangle has sides of 22, 40, and 45 units long. We need to verify whether these lengths meet the following equation. a^2+b^2=c^2 In this equation, c is the length of the longest side. Let's substitute 22, 40, and 45 for a, b, and c respectively. If we get a true statement, the triangle is a right triangle.

We got a false statement, since 2084 is not equal to 2025. Therefore, the triangle is not a right triangle.

We can determine whether △ PQR is a right triangle the same way we did before. a^2+b^2=c^2 The side lengths of △ PQR are sqrt(13), 3, and 2. Notice that the greatest length is sqrt(13), so let's substitute sqrt(13) for c. Also, we will substitute 3 and 2 for a and b.

We got a true statement. This means that △ PQR is a right triangle. As an illustration, let's draw this triangle.

Is the following triple a Pythagorean triple?

(10, 24, 26)

Let's begin by recalling what a Pythagorean triple is.

Pythagorean Triple |- A set of three natural numbers that satisfy the Pythagorean Theorem.

According to the definition, we have to substitute the given numbers into the equation a^2+b^2=c^2 and verify whether we get a true statement. Here, c is the greatest number. Let's write the given triple. (10,24,26) Next, let's substitute 10, 24, and 26 for a, b, and c, respectively.

We got a true statement. Since the three numbers satisfy the Pythagorean Theorem, they form a Pythagorean triple.

Find the value of $x$ for each of the following diagrams. Write exact answers in the simplest form.

We begin by noticing that △ ABD is a right triangle with hypotenuse 5 units long.

We can find the value of x by using the Pythagorean Theorem. a^2+b^2=c^2 However, we do not know the length of BD. On the other hand, BD is one leg of the right triangle BCD. We know two side lengths of this triangle, so we can apply the Pythagorean Theorem to find BD.

Let's substitute 2 for a, BD for b, and sqrt(13) for c into the equation. Then, we will solve the resulting equation for BD.

Now that we know the length of BD, we can use the Pythagorean Theorem once more to find the length of AD, which is x. This time we will substitute 3 and 5 for a and c, respectively.

The length of AD is 4.

The value of x is the length of the hypotenuse of the right triangle ABC. However, we only know one leg length of this triangle, BC=4.

We need to find the length of AC to apply the Pythagorean Theorem and this way find x. a^2+b^2=c^2 Let's focus on polygon ACDE for a moment.

All four sides of ACDE have one hatch mark. This means that all of them are congruent. Since AE=2, we can say that all the side lengths are equal to 2, particularly AC=2. Now we are ready to apply the Pythagorean Theorem to △ ABC and find x.

The length of AB is 2sqrt(5).

For each of the following three-dimensional solids, find the value of $x .$ Write the answers as radicals in their simplest form.

We begin by noticing that △ ABC is a right triangle where ∠ B is the right angle. For this triangle, we know the length of one leg, AB.

We can also see that DE and AC have the same number of hatch marks. This means that both segments have the same length. Therefore, AC=6. Now that we know two side lengths of △ ABC, we can find the value of x by applying the Pythagorean Theorem.

The ramp is 3sqrt(3) units high.

The given cone is 8 units high and the radius of the base is 3 units. We can see that ∠ M is a right angle. This means that △ JMK is a right triangle. We need to find the length of the hypotenuse — the slant height of the cone.

Let's apply the Pythagorean Theorem to find the value of x.

Notice that we cannot simplify sqrt(73) further because 73 is a prime number. We can conclude that the slant height of the given cone is sqrt(73) units long.

This time we are given a pyramid for which we know the lengths of QR and TR. We need to find the length of TQ.

We can see that ∠ QTR is a right angle, which means that △ QTR is a right triangle. Then, we can apply the Pythagorean Theorem to find x.

The length of TQ is 2sqrt(7) units.

Heichi is painting the back of his house. He will need to reach the highest part. He places the foot of a

4 -

meter-long ladder one meter from the wall. How far above the ground is the ladder resting? Round the answer to one decimal place.

Let's begin by making a drawing that illustrates the situation. We know the foot of the ladder is one meter from the wall and the ladder is four meters long.

The ladder and the wall form a right triangle whose hypotenuse is the ladder. We have to find the length of the leg formed at the wall. Let x be the length of this leg. We can find x by applying the Pythagorean Theorem.

The ladder rests at about 3.9 meters from the ground.

What is the distance between the points $A$ and $B ?$ Write the answer in exact form as simplified as possible.

We can find the distance between two points (x_1,y_1) and (x_2,y_2) in the coordinate plane by using the distance formula. d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2) Let's begin by identifying the coordinates of the given points.

Now that we know the coordinates, let's substitute them into the distance formula and evaluate it.

The distance between the points is 2sqrt(17).

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