8th Grade Unit 8 Pythagorean Theorem

The Size of Shapes

In this week your student will be working with the relationship between the side length and area of squares. We know two main ways to find the area of a square:

• Multiply the square’s side length by itself.
• Decompose and rearrange the square so that we can see how many square units are inside. For example, if we decompose and rearrange the tilted square in the diagram, we can see that its area is 10 square units.

But what is the side length of this tilted square? It cannot be 3 units since 32=9 and it cannot be 4 units since 42=16. In order to write “the side length of a square whose area is 10 square units,” we use notation called a square root. We write “the square root of 10” as sqrt10 and it means “the length of a side of a square whose area is 10 square units.” All of these statements are true:

• sqrt9=3 because 32=9
• sqrt16=4 because 42=16
• sqrt10 is the side length of a square whose area is 10 square units, and (\srt10)2=10

Task

If each grid square represents 1 square unit, what is the side length of this titled square? Explain your reasoning.

Solution

The side length is sqrt26 because the area of the square is 26 square units and the square root of the area of a square is the side length.

Side Lengths and Areas of Squares

This week your student will be working with the Pythagorean Theorem, which is useful when working with right triangles. A right triangle is any triangle with a right angle. The side opposite the right angle is called the hypotenuse, and the two other sides are called the legs. Here we have a triangle with hypotenuse c and legs a and b. The Pythagorean Theorem states that for any right triangle, the sum of the squares of the legs are equal to the square of the hypotenuse. In other words, a2+b2=c2.

We can use the Pythagorean Theorem to tell if a triangle is a right triangle or not, to find the value of one side length of a right triangle if we know the other two, and to answer questions about situations that can be modeled with right triangles. For example, let’s say we wanted to find the length of this line segment:

We can first draw a right triangle and determine the lengths of the two legs:

Next, since this is a right triangle, we know 24²+7²=c², which means the length of the line segment, is 25 units.

Tasks

1. Find the length of the hypotenuse as an exact answer using a square root.

2. What is the length of line segment p? Explain or show your reasoning. (Each grid square represents 1 square unit.)

Solutions

1. The length of the hypotenuse is 50−−√ units. With legs a and b both equal to 5 and an unknown value for the hypotenuse, c, we know the relationship 52+52=c2 is true. That means 50=c2, so c must be 50−−√ units.
2. The length of p is 25−−√ or 5 units. If we draw in the right triangle, we have legs of length 3 and 4 and hypotenuse p, so the relationship 32+42=p2 is true. Since 32+42=25=p2, p must equal 25−−√ or 5 units.

Side Lengths and Volumes of Cubes

This week your student will learn about cube roots. We previously learned that a square root is the side length of a square with a certain area. For example, if a square has an area of 16 square units then it’s edge length is 4 units because sqrt16=4. Now, think about a solid cube. The cube has a volume, and the edge length of the cube is called the cube root of its volume. In this diagram, the cube has a volume of 64 cubic units:

Even without the useful grid, we can calculate that the edge length is 4 from the volume since 64−−√3=4.

Cube roots that are not integers (like 4) are still numbers that we can plot on a number line. If we have the three numbers 40−−√, 30−−√3, and 64−−√3, we can plot them on the number line by estimating what integers they are near. For example, 40−−√ is between 6 and 7, since 40−−√>36−−√, which equals 6, and 40−−√<49−−√ and 49−−√=7. Similarly, 40−−√3 is between 3 and 4 because 30 is between 27 and 64. Our number line will look like this:

Task

Plot the given numbers on the number line. 28−−√;27−−√3;50−−√3

Solutions

Since 3³=27 means 27−−√3=3, we plot 27−−√3 at 3. The 50−−√3 is between 3 and 4 because 50−−√3>27−−√3, which equals 3, and 50−−√3<64−−√3, which equals 4. A similar argument can be made for 28−−√ falling between 5 and 6.