The Pythagorean Theorem and the Distance Formula - Geometry in Two Dimensions (Pre-Algebra)

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

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

$LHS = RHS$

650 = c

RearrangeEqn

Rearrange equation

c = 650

Simplify right-hand side

UseCalc

Use a calculator

c = 25.4950 \dots

RoundDec

Round to $1$ decimal place(s)

c \approx 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

b .

Then, solve the equation for the remaining variable.

a^{2} + b^{2} = c^{2}

SubstituteII

$a = 3$ , $c = 5$

3^{2} + b^{2} = 5^{2}

Solve for

b

CalcPow

Calculate power

9 + b^{2} = 25

SubEqn

$LHS - 9 = RHS - 9$

b^{2} = 16

SqrtEqn

$LHS = RHS$

b = 16

CalcRoot

Calculate root

b = 4

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.

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)

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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

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