MAT8 Q2W5D1: Discovering the Pythagorean Theorem

By the end of the lesson, you will be able to:

1. Illustrate and describe the parts of a right triangle (legs, hypotenuse).
2. Explore and observe the relationship between the squares of the sides of a right triangle.
3. Explain in your own words the meaning of the Pythagorean Theorem through a guided activity.

• Right Triangle - a triangle with one angle measuring 90°.
• Legs - the two sides that form the right angle in a right triangle.
• Hypotenuse - the side opposite the right angle, and the longest side of a right triangle.
• Square of a number - the product when a number is multiplied by itself.

Activity: Solve the problem below.

A golf ball has a diameter of 4 cm, and a tennis ball has a diameter of 7 cm. Find the difference between their volumes.

Show Answer

Step 1: Formula for the volume of a sphere is

$V=\frac{4}{3}\pi r^3$

Step 2: Radius of golf ball = 2 cm, radius of tennis ball = 3.5 cm

$V_{\text{golf}}=\frac{4}{3}\pi 2^3$ ≈ 33.51 cm³

$V_{\text{tennis}}=\frac{4}{3}\pi {3.5}^3$ ≈ 179.59 cm³

Step 4: Difference = 179.59 − 33.51 = 146.08 cm³

Part 1: Starting with Shapes You Already Know

Imagine standing in the corner of a basketball court. You see the baseline forming one line, and the sideline forming another. At that corner, the two lines meet to form a perfect square corner. That kind of corner is called a right angle.

A triangle that has one right angle (90°) is called a right triangle.

Guiding Question: What do we call the triangle that contains a right angle?

Show Answer

It is called a right triangle.

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Part 2: Meeting the Right Triangle

A right triangle has three important parts:

• Two legs - the sides that form the right angle.
• One hypotenuse - the side opposite the right angle, and always the longest side.

Let’s label them as follows: the legs are a and b, and the hypotenuse is c.

Checkpoint: Which side of a right triangle is always the longest?

Show Answer

The hypotenuse.

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Part 3: A Curious Relationship

In mathematics, sometimes we discover that numbers or shapes have a surprising relationship. One of the most famous is the Pythagorean Theorem.

Activity:

1. Draw a right triangle on graph paper where one leg is 3 units and the other leg is 4 units.
2. On each leg, draw a square using the leg as a side.
3. On the hypotenuse, draw a square using it as a side.

Measure the areas of the three squares.

Guiding Questions:

1. What is the area of the square on the 3-unit side?
2. What is the area of the square on the 4-unit side?
3. What is the area of the square on the hypotenuse?
4. What do you notice when you compare them?

Show Answer

1. 9 square units
2. 16 square units
3. 25 square units
4. The sum of the areas of the smaller two squares equals the area of the largest square.

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Part 4: Stating the Theorem

From that discovery, mathematicians thousands of years ago concluded a rule:

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

This is called the Pythagorean Theorem, named after the Greek philosopher and mathematician Pythagoras.

Mini-Summary: The theorem tells us that the square of the hypotenuse equals the sum of the squares of the two legs.

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Part 5: Real-World Connections

• Construction - Carpenters use the Pythagorean Theorem to check if walls meet at a perfect right angle by measuring diagonals.
• Sports - In soccer or basketball, a diagonal run distance can be found using the theorem.
• Navigation - A pilot flying north and then east can compute the straight-line distance back to the starting point using this theorem.
• Design - Architects and engineers use it to calculate supports, ramps, and ladders safely.

Checkpoint: If a ladder is 10 m long and placed 6 m away from the wall, how high does it reach?

Show Answer

$b=\sqrt{c^2-a^2}$

$b=\sqrt{100-36}$

$b=\sqrt{64}$

b = 8 m

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Part 6: Historical Note

The Pythagorean Theorem has been known for thousands of years. Ancient civilizations like the Babylonians and Egyptians used it long before Pythagoras was born. Pythagoras is credited because he provided a logical proof. There are over 370 different proofs of this theorem, including one by U.S. President James Garfield.

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Part 7: Applying Step by Step

Example: Find the hypotenuse of a right triangle if the legs are 5 cm and 12 cm.

Show Answer

$c^2=5^2+{12}^2$

$c^2=25+144$

$c^2=169$

c = 13 cm

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Part 8: Your Turn

1. A right triangle has one leg of 9 cm and another of 12 cm. Find the hypotenuse.
2. A ladder 15 ft long rests against a wall, touching the ground 9 ft away. How high does it reach?

Show Answer

1. 15 cm
2. 12 ft

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Mini-Summary of Day 1 Explore

• A right triangle has legs and a hypotenuse.
• The Pythagorean Theorem states: $c^2=a^2+b^2$
• Real-life problems in sports, construction, and navigation use this theorem.
• Day 1 focused on discovering and exploring the relationship between the squares of the sides.

References

• Intro to the Pythagorean theorem. (n.d.). [Video]. Khan Academy. https://www.khanacademy.org/math/geometry-home/geometry-pythagorean-theorem/pythagorean-theorem-basic-geo/v/the-pythagorean-theorem
• TED-Ed. (2017, September 11). How many ways are there to prove the Pythagorean theorem? - Betty Fei [Video]. YouTube. https://www.youtube.com/watch?v=YompsDlEdtc
• Burton, D. M. (2011). The History of Mathematics: An Introduction. McGraw-Hill.
• Maor, E. (2007). The Pythagorean Theorem: A 4,000-Year History. Princeton University Press.

Worked Example 1

Find the length of the hypotenuse of a right triangle with legs 6 cm and 8 cm.

Show Answer

Step 1: Recall the Pythagorean Theorem.

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

Step 2: Substitute values.

$c^2=6^2+8^2$

Step 3: Simplify.

$c^2=36+64$

$c^2=100$

Step 4: Take the square root.

c = 10 cm

Worked Example 2

The hypotenuse of a right triangle is 13 m, and one leg is 5 m. Find the other leg.

Show Answer

Step 1: Formula:

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

Step 2: Substitute values.

${13}^2=a^2+5^2$

Step 3: Simplify.

$169=a^2+25$

Step 4: Solve for a².

$a^2=144$

Step 5: Take square root.

a = 12 m

Worked Example 3

A ladder is 15 ft long and placed 9 ft away from a wall. How high does it reach?

Show Answer

Step 1: Identify sides.

• a = 9 ft (distance from wall)
• c = 15 ft (ladder)
• b = ? (height up wall)

Step 2: Formula:

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

Step 3: Substitute.

${15}^2=9^2+b^2$

$225=81+b^2$

Step 4: Solve.

$b^2=144$

b = 12 ft

Worked Example 4

Find the diagonal of a rectangle that measures 9 m by 12 m.

Show Answer

Step 1: Recognize it forms a right triangle with legs 9 and 12.

Step 2: Apply theorem.

$c^2=9^2+{12}^2$

$c^2=81+144$

$c^2=225$

c = 15 m

Worked Example 5

A computer screen is 24 inches wide and 18 inches tall. What is the diagonal screen size?

Show Answer

Step 1: Recognize rectangle - right triangle.

Step 2: Apply theorem.

$c^2={24}^2+{18}^2$

$c^2=576+324$

$c^2=900$

c = 30 inches

Now You Try

1. Find the hypotenuse if the legs are 7 cm and 24 cm.
2. A right triangle has a hypotenuse of 25 m and one leg of 20 m. Find the other leg.
3. A 40-inch TV has a width of 32 inches. How tall is it?
4. A ramp is 12 m long and rises 5 m high. How far is its base from the wall?
5. A soccer field is 100 m long and 60 m wide. What is the diagonal distance from one corner to the opposite?

Show Answer

1. 25 cm
2. 15 m
3. 24 inches
4. √119 ≈ 10.9 m
5. √13600 = 116.6 m (approx.)

1. Find the hypotenuse of a right triangle whose legs are 8 cm and 15 cm.
   Show Answer

   $c^2=8^2+{15}^2$

   $c^2=64+225$

   $c^2=289$

   c = 17 cm

2. The hypotenuse of a right triangle is 29 m and one leg is 20 m. Find the other leg.
   Show Answer

   ${29}^2=a^2+{20}^2$

   $841=a^2+400$

   $a^2=441$

   a = 21 m

3. A ladder is 25 ft long and reaches 24 ft up a wall. How far is its base from the wall?
   Show Answer

   $a^2={25}^2-{24}^2$

   $a^2=625-576$

   $a^2=49$

   a = 7 ft

4. A TV has a height of 45 cm and width of 80 cm. What is its diagonal?
   Show Answer

   Diagonal = $\sqrt{{45}^2+{80}^2}$ ≈ 91.8 cm

5. A soccer field measures 120 m by 90 m. Find the diagonal distance across it.
   Show Answer

   Diagonal = $\sqrt{{120}^2+{90}^2}$ = 150 m

6. A kite string is 50 m long and is tied 14 m above the ground on a pole. How far from the base of the pole is the kite?
   Show Answer

   Distance = $\sqrt{{50}^2-{14}^2}$ = 48 m

7. The base of a triangle is 9 cm and the hypotenuse is 15 cm. Find the missing leg.
   Show Answer

   b² = 225 − 81 = 144; b = 12 cm

8. A phone screen is 5.5 inches wide and 7 inches tall. Find its diagonal size.
   Show Answer

   Diagonal = √79.25 ≈ 8.9 inches

9. Find the missing leg of a right triangle if the hypotenuse is 41 cm and one leg is 40 cm.
   Show Answer

   Missing leg = √81 = 9 cm

10. A rectangular billboard is 12 m tall and 16 m wide. What is the diagonal measurement?
    Show Answer

    Diagonal = √400 = 20 m

1. What makes a triangle a right triangle?
   Show Answer

   A right triangle contains one 90° angle.

2. In a right triangle, which side is always the longest?
   Show Answer

   The hypotenuse.

3. The legs of a right triangle measure 9 cm and 12 cm. Find the hypotenuse.
   Show Answer

   $c^2=9^2+{12}^2$ = 225; c = 15 cm

4. The hypotenuse of a right triangle is 25 m, and one leg is 7 m. Find the other leg.
   Show Answer

   a² = 625 − 49 = 576; a = 24 m

5. State the Pythagorean Theorem using variables.
   Show Answer

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

6. A ladder is leaning against a wall. The base is 9 ft from the wall, and the ladder is 15 ft long. How high does it reach?
   Show Answer

   b² = 225 − 81 = 144; b = 12 ft

7. A rectangle has length 20 m and width 21 m. Find the diagonal.
   Show Answer

   Diagonal = √841 = 29 m

8. A right triangle has sides 8 m, 15 m, and 17 m. Is it a right triangle?
   Show Answer

   $8^2$ + ${15}^2$ = $64+225=289$ ; ${17}^2=289$ . Since equal, yes, it is a right triangle.

9. A computer monitor is 16 in wide and 12 in tall. Find the diagonal size.
   Show Answer

   Diagonal = √400 = 20 in

10. A ship sails 30 km north and then 40 km east. How far is it from its starting point?
    Show Answer

    Distance = √2500 = 50 km

11. Find the missing leg of a right triangle if the hypotenuse is 50 and the other leg is 14.
    Show Answer

    Missing leg = √2304 = 48

12. A rectangular billboard is 9 m tall and 40 m wide. What is the diagonal length?
    Show Answer

    Diagonal = √1681 = 41 m

13. If c² = a² + b², what type of triangle is it?
    Show Answer

    It is a right triangle.

14. A rectangular garden has dimensions 24 m by 32 m. What is the diagonal path across the garden?
    Show Answer

    Diagonal = √1600 = 40 m

15. A right triangle has one leg 36 m and hypotenuse 85 m. Find the other leg.
    Show Answer

    Missing leg = √5929 = 77 m

1. Ancient Builders’ Secret: Egyptian builders used a rope with 12 equally spaced knots to form right angles when building pyramids.
   • If one side of the triangle measured 3 units and another side 4 units, what was the hypotenuse?
   • Why do you think they chose this “3-4-5” triangle for construction?
   Show Answer

   Hypotenuse = 5 units

   They chose it because it always guarantees a perfect right angle without needing advanced tools.

2. Designing a Ramp: A wheelchair ramp must be built to rise 1.5 m while extending 6 m along the ground.
   • What is the actual length of the ramp?
   • Why must engineers know this before building?
   Show Answer

   Ramp length = √38.25 ≈ 6.19 m

   Engineers must know this to ensure the ramp’s slope is safe and accessible.

3. GPS Shortcuts: A delivery rider travels 2 km east and then 3 km north.
   • How far is the rider from the starting point in a straight line?
   • If they must return, why is this distance useful to know?
   Show Answer

   Distance = √13 ≈ 3.6 km

   Knowing this helps the rider plan shortest routes and estimate fuel use.

4. Sports Strategy: In basketball, a player runs baseline 8 m and then sideline 6 m. Another player runs diagonally to the hoop.
   • Who runs the shorter distance?
   • By how many meters?
   Show Answer

   Diagonal = √100 = 10 m

   Baseline + sideline = 14 m. The diagonal is 4 m shorter.

5. Creative Challenge - Build Your Own Problem: Think of a real situation (at home, in school, or in the community) where the Pythagorean Theorem could help solve a problem. Write it down and solve it using the formula.
   Show Answer

   Example: A fireman’s ladder is 20 m long. It is placed 12 m from the base of a building. Height reached = √256 = 16 m.

Take a few minutes to reflect on today’s learning. Write your answers in your notebook.

Reflection Task (3-2-1):

• 3 things you learned today about right triangles and the Pythagorean Theorem.
• 2 questions you still have or want to explore more deeply.
• 1 real-life situation where you think you can apply the Pythagorean Theorem.