Finish this study guide 2.1.1Study: The Pythagorean Theorem

Study Guide

Mathematics Grade 8

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Main idea: Proving the Pythagorean theorem
1. Fill in the blanks and circle the correct words to describe the Pythagorean theorem.
The Pythagorean theorem relates the of a triangle.
It states that the square of the length of the [longest/shortest] side is equal to the [sum/difference] of the squares of the lengths of the other two sides.
2. Watch the animation of the first geometric proof of the Pythagorean theorem. Then use the diagram to complete the sentences.

The areas of figure 1 and figure 2 above are [equal/not equal].
Figures 1 and 2 both contain right triangles with side lengths a, b, and c.
After subtracting the areas of the triangles from each figure, the remaining areas of figure 1 and figure 2 must still be [equal/not equal].
The area of the square with side length a is a2. The area of the square with side length b is . The area of the square with side length c is .
Write the Pythagorean theorem in the blanks below the diagram.

Pythagorean theorem: + =
3. Watch the animation of the second proof of the Pythagorean theorem. Then fill in the blanks.
Two squares are placed side by side. The area of the combined shape is the sum of the squares' areas: + .
This combined shape is then rearranged to form a square with an area of .

The combined shape has the same area as the resulting square, so a2___ b2___ c2.

Main idea: Finding unknown side lengths in right triangles
4. Answer the question and fill in the blanks to describe right triangles.
What is a right triangle?
The hypotenuse of a triangle is always the side. Its length is always represented by the letter in the Pythagorean theorem.
Label the parts of the right triangle below. Use the terms right angle, hypotenuse, and leg. One term will be used twice.

5. Complete the calculations to find the length of side c of the right triangle.

6. Use the Pythagorean theorem to find the length of side b of the right triangle.

7. Complete the sentence.
Finding the length of the of a rectangle is the same as finding the length of the hypotenuse of each right triangle that makes up the rectangle.
Main idea: Finding unknown lengths in three-dimensional objects
8. Fill in the blanks to describe right triangles inside right pyramids.
The vertical leg of a triangle within a right pyramid is the of the pyramid.
The other leg is equal to the length of a side of the pyramid's base, b.
The hypotenuse is the , s, of the pyramid.
9. Complete the steps to find the slant height, s, of the triangle in the right pyramid.

A side of the base of the pyramid is m.

That means side a of the triangle must equal m.
Using the Pythagorean theorem:

Take the square root to find that the slant height, s, of the triangle is m.
10. Label the parts of the right cone in the diagram. Use the terms radius, height, and slant height.

11. The radius of a right cone is 8 inches. The slant height of the cone is 10 inches. Use the Pythagorean theorem to find the height of the cone.

Main idea: Using the Pythagorean theorem in real-world examples
12. Fill in the blanks to describe how to find the dimensions of a TV.
The size of a TV is described by the length of its screen's diagonal. This is the same as the length of the of one of the right triangles that makes up the TV screen.
The Pythagorean theorem for finding the diagonal length of a TV could be written as: (height of TV)2 + ( of TV)2 = (diagonal of TV)2
13. Use the diagram to answer the questions about Shana's pencil holder.

Which side represents the height of the right triangle in the diagram?
Which side represents the hypotenuse of the right triangle?
What does the hypotenuse represent in the context of the problem?

Side a is 6 inches long and side b is 8 inches long, so side c must be inches long. What does this mean in the context of the problem?