Pythagorean Theorem

Relationships in right triangles

Pythagorean Theorem

The Theorem

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs.

$a^2 + b^2 = c^2$

Where:

$a$ and $b$ are the legs
$c$ is the hypotenuse (longest side, opposite the right angle)

Using the Theorem

To find the hypotenuse: $c = \sqrt{a^2 + b^2}$

To find a leg: $a = \sqrt{c^2 - b^2}$

Pythagorean Triples

Sets of three positive integers that satisfy $a^2 + b^2 = c^2$ :

Common triples:

$3, 4, 5$
$5, 12, 13$
$8, 15, 17$
$7, 24, 25$

Any multiple works too: $6, 8, 10$ (multiply $3, 4, 5$ by 2)

Converse

If $a^2 + b^2 = c^2$ , then the triangle is a right triangle.

Distance Formula

The Pythagorean Theorem leads to the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

📚 Practice Problems

1Problem 1easy

❓ Question:

A right triangle has legs of length 3 and 4. Find the length of the hypotenuse.

💡 Show Solution

Step 1: Recall the Pythagorean Theorem: a² + b² = c² where a and b are legs, c is the hypotenuse

Step 2: Identify the given values: a = 3, b = 4, c = ?

Step 3: Substitute into the formula: 3² + 4² = c² 9 + 16 = c² 25 = c²

Step 4: Solve for c: c = √25 c = 5

Step 5: Verify: 3² + 4² = 9 + 16 = 25 = 5² ✓ This is a 3-4-5 Pythagorean triple

Answer: The hypotenuse is 5

2Problem 2easy

❓ Question:

Find the length of the hypotenuse if the legs of a right triangle are 3 and 4.

💡 Show Solution

Use $a^2 + b^2 = c^2$ :

$3^2 + 4^2 = c^2$

$9 + 16 = c^2$

$25 = c^2$

$c = 5$

Answer: The hypotenuse is $5$

3Problem 3easy

❓ Question:

A right triangle has a hypotenuse of 13 and one leg of length 5. Find the other leg.

💡 Show Solution

Step 1: Use the Pythagorean Theorem: a² + b² = c²

Step 2: Substitute known values: 5² + b² = 13² 25 + b² = 169

Step 3: Solve for b²: b² = 169 - 25 b² = 144

Step 4: Solve for b: b = √144 b = 12

Step 5: Verify: 5² + 12² = 25 + 144 = 169 = 13² ✓ This is a 5-12-13 Pythagorean triple

Answer: The other leg is 12

4Problem 4medium

❓ Question:

A right triangle has a hypotenuse of 13 and one leg of 5. Find the other leg.

💡 Show Solution

Use $a^2 + b^2 = c^2$ :

$5^2 + b^2 = 13^2$

$25 + b^2 = 169$

$b^2 = 144$

$b = 12$

Answer: The other leg is $12$

5Problem 5medium

❓ Question:

A ladder is leaning against a wall. The ladder is 10 feet long and the base is 6 feet from the wall. How high up the wall does the ladder reach?

💡 Show Solution

Step 1: Visualize the right triangle:

Ladder = hypotenuse = 10 feet
Distance from wall = one leg = 6 feet
Height on wall = other leg = ?

Step 2: Use Pythagorean Theorem: 6² + h² = 10²

Step 3: Substitute and solve: 36 + h² = 100 h² = 100 - 36 h² = 64

Step 4: Find h: h = √64 h = 8

Step 5: Verify: 6² + 8² = 36 + 64 = 100 = 10² ✓ This is a 6-8-10 triangle (multiple of 3-4-5)

Answer: The ladder reaches 8 feet up the wall

6Problem 6medium

❓ Question:

Find the length of the diagonal of a rectangle with length 15 cm and width 8 cm.

💡 Show Solution

Step 1: Recognize the right triangle: The diagonal of a rectangle creates a right triangle with the length and width as legs

Step 2: Use Pythagorean Theorem: a² + b² = c² where a = 8, b = 15, c = diagonal

Step 3: Substitute: 8² + 15² = c² 64 + 225 = c² 289 = c²

Step 4: Solve for c: c = √289 c = 17

Step 5: Verify: 8² + 15² = 64 + 225 = 289 = 17² ✓ This is an 8-15-17 Pythagorean triple

Answer: The diagonal is 17 cm

7Problem 7hard

❓ Question:

Is a triangle with sides 7, 24, and 25 a right triangle?

💡 Show Solution

Use the converse of the Pythagorean Theorem.

Check if $a^2 + b^2 = c^2$ (where $c = 25$ is the longest side):

$7^2 + 24^2 = 49 + 576 = 625$

$25^2 = 625$

Since $7^2 + 24^2 = 25^2$ , the triangle is a right triangle.

Answer: Yes, it is a right triangle (and $7, 24, 25$ is a Pythagorean triple)

8Problem 8hard

❓ Question:

A baseball diamond is a square with sides of 90 feet. What is the distance from home plate to second base (the diagonal of the square)?

💡 Show Solution

Step 1: Visualize the problem: A square with side = 90 feet Diagonal connects home plate to second base Diagonal splits square into two right triangles

Step 2: Use Pythagorean Theorem: Both legs of the right triangle = 90 feet 90² + 90² = d²

Step 3: Calculate: 8100 + 8100 = d² 16200 = d²

Step 4: Solve for d: d = √16200

Step 5: Simplify the radical: 16200 = 8100 × 2 = 90² × 2 d = √(90² × 2) d = 90√2

Step 6: Calculate decimal approximation: d ≈ 90 × 1.414 d ≈ 127.3 feet

Step 7: Alternative recognition: This is a 45-45-90 triangle If leg = 90, then hypotenuse = 90√2 ✓

Answer: The distance is 90√2 feet (approximately 127.3 feet)

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