The Pythagorean Theorem

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Examples, solutions, and videos to help Grade 8 students learn that when the square of a side of a right triangle represented as a², b² or c² is not a perfect square, they can estimate the side length as between two integers and identify the integer to which the length is closest.

New York State Common Core Math Grade 8, Module 7, Lesson 1

Worksheets for Grade 8

Lesson 1 Summary

Perfect square numbers are those that are a product of an integer factor multiplied by itself. For example, the number 25 is a perfect square number because it is the product of multiplied 5 by 5.

When the square of the length of an unknown side of a right triangle is not equal to a perfect square, you can estimate the length by determining which two perfect squares the number is between.

Lesson 1 Classwork

Show students the three triangles below.
The first triangle requires students to use the Pythagorean Theorem to determine that the unknown side length.
The second triangle requires students to use the converse of the theorem to determine that it is a right triangle.
The third triangle requires students to use the converse of the theorem to determine that it is not a right triangle.

Recall the Pythagorean Theorem and its converse for right triangles.
• The Pythagorean Theorem states that a right triangle with leg lengths a and b and hypotenuse c will satisfy a² + b² = c².
The converse of the theorem states that if a triangle with side lengths a, b, and c satisfy the equation a² + b² = c², then the triangle is a right triangle.

Example 1 - Example 3
Write an equation that will allow you to determine the length of the unknown side of the right triangle.
Example 4
In the figure below, we have an equilateral triangle with a height of 10 inches. What do we know about an equilateral triangle?
Exercises 1–3
1. Use the Pythagorean Theorem to estimate the length of the unknown side of the right triangle. Explain why your estimate makes sense.

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