WEEK 31-RIGHT-TRIANGLES-AND-THE-PYTHAGOREAN-THEOREM

Objectives

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Right Triangles and The Pythagorean Theorem

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Story

Right Triangles

Exercises and Applications

Summary

The Pythagorean Theorem

Story

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Right Triangles

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What is a Right Triangle?

Definition: A triangle with one angle that measures exactly 90° (a right angle)

Example 1: This is a right triangle:

These sides are called legs (where is formed the right angle)

Hypothenuse: Is the largest side of the triangle.

The Pythagoream Theorem

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Theorem: The sum of the squared legs of a right triangle equals to the hypothenuse squared.

a²+b²=c²

are the lenght of the sides.

is the hypothenuse lenght .

a b

c

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Pythagorean theorem states the relationship between the lengths of the sides of a right triangle.

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b²

a²

c²

Is it a Right Triangle?

Property: If , then the triangle with sides is a right triangle.

Example 1: Determine if a triangle with sides 6, 8, and 10 is a right triangle.

The property is verified

a²+b²=c²

a, b, and c

a=3, b=4, c=5

1. Assign letters to the givens.

2.

2. Replace the given values into the Pythagorean theorem: a²+b²=c²

3. If the solution is the same in both sides it is always true.

Let's see some examples:

Example 1: If one side a = 0.6 units and the hypotenuse c = 1 units, find b.

Example 2: Find the hypothenuse in the triangle shown below.

c=13

a=0.6, b=?, c=1

a= 0.8 units

a=5, b=12, c=?

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1. Assign letters to the givens and the unknown.

2. Replace the given values into the Pythagorean theorem: a²+b²=c²

3. Calculate square numbers and isolate the unknown b.

4. Apply square root.

2. Replace the given values into the Pythagorean theorem: a²+b²=c²

3. Calculate square numbers in order to find c.

4. Apply square root.

1.

Exercises and Applications

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Reaching the Roof with a Ladder

A ladder leaning against a wall forms a right triangle. If the ladder is 10 meter long and the bottom of the ladder is 6 meter from the wall,

how high does the ladder reach?

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Solution

a=6, b=?, c=10

a=8 meter

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1. Create a diagram of the situation. Put the given and question on it.

1. Assign letters to the givens.

2. Replace the given values into the Pythagorean theorem: a²+b²=c²

3. Calculate square numbers and isolate the unknown a.

4. Apply square root.

Data follows a positive association.

Determine if the three lenghts can form a right triangle or not.

3, 4, 5

2, 2, 3

5,6,7

5,12,13

Yes

No

• 3, 4, 5 -----> Yes
• 2, 2, 3 -----> No
• 5,6,7 -----> No
• 5,12,13 -----> Yes

The solution will appear in 60 seconds

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Summary: Right Triangles

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a leg: One of the sides forming the right angle.

Hypothenuse: The largest side of the triangle.

b leg: One of the sides forming the right angle.

Pythagorean theorem: c²=a²+b²

Welcome 6th graders!

A journey soon begin through Social Science experiences!

Great job!

See you next time

8TH-RIGHT-TRIANGLES-AND-THE-PYTHAGOREAN-THEOREM-EN © 2024 by CASURID is licensed under CC BY-NC-ND 4.0

It is highly advised to have:

MATERIAL

• Grid paper.
• A Protactor.
• Pencils of different colors.
• Eraser.
• A rule.

• A compass.
• A calculator.
• Geogebra installed on your phone/tablet/computer (or use online version).

"MA.8.GR.1 Develop an understanding of the Pythagorean Theorem and angle relationships involving triangles."MA.8.GR.1.1 Apply the Pythagorean Theorem to solve mathematical and real-world problems involving unknown side lengths in right triangles.MA.8.NSO.1 Solve problems involving rational numbers, including numbers in scientific notation, and extend the understanding of rational numbers to irrational numbers.MA.8.NSO.1.7 Solve multi-step mathematical and real-world problems involving the order of operations with rational numbers including exponents and radicals.MA.K12.MTR.7.1 Apply mathematics to real-world contexts.