Unit 8 Pythagorean Theorem And Irrational Numbers — Unit Plan

• $\sqrt{9}=3$ because $3^2=9$.

• $\sqrt{16}=4$ because $4^2=16$.

The value of $\sqrt{10}$ must be between 3 units and 4 units because it is between the values of $\sqrt{9}$ and $\sqrt{16}$.

There are 3 squares on a square grid, arranged in order of area, from smallest, on the left, to largest, on the right. The left most square is aligned to the grid and has side lengths of 3 with an area of 9. The middle square is tilted on the grid so that its sides are diagonal to the grid. The square is labeled with a side length of square root of 10 and an area of 10. The right most square is aligned to the grid and has side lengths of 4 with an area of 16.

A square whose area is 25 square units has a side length of $\sqrt{25}$ units, which means that $\sqrt{25} \boldcdot \sqrt{25} = 25$. Since $5 \boldcdot 5 = 25$, we know that $\sqrt{25}=5$.

$\sqrt{25}$ is an example of a rational number. A rational number is a fraction or its opposite. In an earlier grade we learned that $\frac{a}{b}$ is a point on the number line found by dividing the interval from 0 to 1 into $b$ equal parts and finding the point that is $a$ of them to the right of 0. We can always write a fraction in the form $\frac{a}{b}$, where $a$ and $b$ are integers (and $b$ is not 0), but there are other ways to write them. For example, we can write $\sqrt{25}=\frac51=5$ or $\text-\frac{1}{\sqrt{4}} = \text-\frac{1}{2}$. Because fractions and ratios are closely related ideas, fractions and their opposites are called rational numbers.

Now consider a square whose area is 2 square units with a side length of $\sqrt{2}$ units. This means that$\sqrt{2} \boldcdot \sqrt{2} = 2$.

An irrational number is a number that is not rational, meaning it cannot be expressed as a positive or negative fraction. For example,
$\sqrt{2}$ has a location on the number line (it’s a tiny bit to the right of $\frac75$),
but its location can not be found by dividing the segment from 0 to 1 into $b$ equal parts and going $a$ of those parts away from 0.

A number line with 10 evenly spaced tick marks. The first tick mark is labeled 0 and the sixth tick mark is labeled 1. An arrow points to the eighth tick mark and is labeled seven-fifths. A second arrow points to a point slightly to the right of the eighth tick mark and is labeled the square root of 2.

$\frac{17}{12}$ is close to $\sqrt{2}$ because $\left( \frac{17}{12} \right)^2=\frac{289}{144}$, which is very close to 2 since $\frac{288}{144}=2$. We could keep looking forever for rational numbers that are solutions to $x^2=2$, and we would not find any since $\sqrt{2}$ is an irrational number.

The square root of any whole number is either a whole number, like $\sqrt{36}=6$ or $\sqrt{64}=8$, or an irrational number. Here are some examples of irrational numbers: $\sqrt{10}, \text{ -}\sqrt3, \text{ }\frac{\sqrt5}{2},\text{ } \pi$.

Here is a line segment on a grid. How can we determine the length of this line segment?

By drawing some circles, we can tell that it’s longer than 2 units, but shorter than 3 units.

Two circles that have the same center are drawn on a square grid with radii 2 and 3. A line segment slanted upward and to the left is drawn such that the bottom endpoint is the center of the two circles and is 1 unit down and 2 units right of the top endpoint of the line segment.

To find an exact value for the length of the segment, we can build a square on it, using the segment as one of the sides of the square.

The area of this square is 5 square units. That means the exact value of the length of its side is $\sqrt5$ units.

Notice that 5 is greater than 4, but less than 9. That means that $\sqrt5$ is greater than 2, but less than 3. This makes sense because we already saw that the length of the segment is in between 2 and 3.

With some arithmetic, we can get an even more precise idea of where $\sqrt5$ is on the number line. The image with the circles shows that $\sqrt5$ is closer to 2 than 3, so let’s find the value of 2.1² and 2.2² and see how close they are to 5. It turns out that $2.1^2=4.41$ and $2.2^2=4.84$, so we need to try a larger number. If we increase our search by a tenth, we find that $2.3^2=5.29$. This means that $\sqrt5$ is greater than 2.2, but less than 2.3. If we wanted to keep going, we could try $2.25^2$ and eventually narrow the value of $\sqrt5$ to the hundredths place. Calculators do this same process to many decimal places, giving an approximation like $\sqrt5 \approx 2.2360679775$. Even though this is a lot of decimal places, it is still not exact because $\sqrt5$ is irrational.

In general, we can approximate the value of a square root by observing the whole numbers around it and remembering the relationship between square roots and squares. Here are some examples:

• $\sqrt{65}$ is a little more than 8 because $\sqrt{65}$ is a little more than $\sqrt{64}$, and $\sqrt{64}=8$.
• $\sqrt{80}$ is a little less than 9 because $\sqrt{80}$ is a little less than $\sqrt{81}$, and $\sqrt{81}=9$.
• $\sqrt{75}$ is between 8 and 9 (it’s 8 point something) because 75 is between 64 and 81.
• $\sqrt{75}$ is approximately 8.67 because $8.67^2=75.1689$.

A number line with the numbers 8 through 9, in increments of zero point 1, are indicated. The square root of 64 is indicated at 8. The square root of 65 is indicated between 8 and 8 point 1, where the square root of 65 is closer to 8 point 1. The square root of 75 is indicated between 8 point 6 and 8 point 7, the square root of 75 is closer to 8 point 7. The square root of 80 is indicated between 8 point 9 and 9, where the square root of 80 is closer to 8 point 9. The square root of 81 is indicated at 9.

If we want to find the square root of a number between two whole numbers, we can work in the other direction. For example, since $22^2 = 484$ and $23^2 = 529$, then we know that $\sqrt{500}$ (to pick one possibility) is between 22 and 23. Many calculators have a square root command, which makes it simple to find an approximate value of a square root.

A right triangle is a triangle with a right angle. In a right triangle, the side opposite the right angle is called the hypotenuse, and the two other sides that make the right angle are called its legs.

Here are some right triangles with the hypotenuse and legs labeled:

If the triangle is a right triangle, then $a$ and $b$ are used to represent the lengths of the legs, and $c$ is used to represent the length of the hypotenuse. The hypotenuse is always the longest side of a right triangle. 

Here are some other right triangles:

Three right triangles are indicated. A square is drawn using each side of the triangles. The triangle on the left has the square labels “a squared equals 16” and “b squared equals 9” attached to each of the legs. The square labeled “c squared equals 25” is attached to the hypotenuse. The triangle in the middle has the square labels “a squared equals 16” and “b squared equals 1” attached to each of the legs. The square labeled “c squared equals 17” is attached to the hypotenuse. The triangle on the right has the square labels “a squared equals 9” and “b squared equals 9” attached to each of the legs. The square labeled “c squared equals 18” is attached to the hypotenuse.

Notice that for these examples of right triangles, the square of the hypotenuse is equal to the sum of the squares of the legs. In the first right triangle in the diagram, $16+9=25$, in the second, $16+1=17$, and in the third, $9+9=18$. Expressed another way, we have:

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

This is a property of all right triangles, not just these examples, and is often known as the Pythagorean Theorem. The name comes from a mathematician named Pythagoras who lived in ancient Greece around 2,500 BCE, but this property of right triangles was also discovered independently by mathematicians in other ancient cultures including Babylon, India, and China. In China, a name for the same relationship is the Shang Gao Theorem.

It is important to note that this relationship does not hold for all triangles. Here are some triangles that are not right triangles. Notice that the lengths of their sides do not have the special relationship $a^2+b^2=c^2$. That is, $16+10$ does not equal 18, and $10+2$ does not equal 16.

Two right triangles are indicated. A square is drawn using each side of the triangles. The triangle on the left has the square labels “a squared equals 16” aligned to the bottom horizontal leg and “b squared equals 10” aligned to the left leg. The square labeled “c squared equals 18 is aligned with the hypotenuse. The triangle on the right has the square labels of “a squared equals 10” aligned with the bottom leg and “b squared equals 2” aligned with the left leg. The square labeled “c squared equals 16” is aligned with the hypotenuse.

The figures shown can be used to see why the Pythagorean Theorem is true. Both large squares have the same area, but they are broken up in different ways. When the sum of the four areas in Square F is set equal to the sum of the 5 areas in Square G, the result is $a^2+b^2=c^2$, where $c$ is the hypotenuse of the triangles in Square G and also the side length of the square in the middle. 

First of two squares of the same area. This square is divided into the following: A square with side lengths “a”. Two rectangles with side lengths “a” and “b”. A square with side lengths “b”.

Second of two squares of the same area. This square is divided into the following: Four identical triangles on each corner of the square with sides labeled “a” and “b”. A square in the center with unlabeled side lengths.

This is true for any right triangle. If the legs are $a$ and $b$ and the hypotenuse is $c$, then $a^2+b^2=c^2$.

For example, to find the length of side $c$ in this right triangle, we know that $24^2+7^2=c^2$. The solution to this equation (and the length of the side) is $c=25$.

The Pythagorean Theorem can be used to find an unknown side length in a right triangle as long as the length of the other two sides is known.

For example, here is a right triangle, where one leg has a length of 5 units, the hypotenuse has a length of 10 units, and the length of the other leg is represented by $g$.

Start with $a^2+b^2=c^2$, make substitutions, and solve for the unknown value. Remember that $c$ represents the hypotenuse, the side opposite the right angle. For this triangle, the hypotenuse is 10.

$\begin{aligned} a^2+b^2&=c^2 \\ 5^2+g^2&=10^2 \\ g^2&=10^2-5^2 \\ g^2&=100-25 \\ g^2&=75 \\ g&=\sqrt{75} \\ \end{aligned}$

Use estimation strategies to know that the length of the other leg is between 8 and 9 units, since 75 is between 64 and 81. A calculator with a square root function gives $\sqrt{75} \approx 8.66$.

The Pythagorean Theorem can be used to solve any problem that can be modeled with a right triangle where the lengths of two sides are known and the length of the other side needs to be found.

For example, let’s say a cable is being placed on level ground to support a tower. It’s a 17-foot cable, and the cable should be connected 15 feet up the tower. How far away from the bottom of the tower should the other end of the cable connect to the ground?

It is often very helpful to draw a diagram of a situation, such as the one shown here:

A right triangle, with the vertical leg labeled tower has a length of 15 units. The hypotenuse, labeled cable has a length of 17 units. The length of the other leg is horizontal and represented by the letter a.

It’s assumed that the tower makes a right angle with the ground. Since this is a right triangle, the relationship between its sides is $a^2+b^2=c^2$, where $c$ represents the length of the hypotenuse and $a$ and $b$ represent the lengths of the other two sides. The hypotenuse is the side opposite the right angle. Making substitutions gives $a^2+15^2=17^2$. Solving this for $a$ gives $a=8$. So the other end of the cable should connect to the ground 8 feet away from the bottom of the tower.

We can use the Pythagorean Theorem to find the distance between any two points in the coordinate plane.

For example, if the coordinates of point $A$ are $(\text-2,\text-3)$, and the coordinates of point $B$ are $(\text-8,4)$, the distance between them is also the length of line segment $AB$. It is a good idea to plot the points first.

The graph of a line segment in the coordinate plane with the origin labeled “O”. On the x-axis, the numbers negative 8 through 0 are indicated. On the y-axis, the numbers negative 3 through 4 are indicated. The line segment begins to the left of the y axis and above the x axis at the point labeled B where point B has coordinates negative 8 comma 4. The line segment slants downward and to the right, crosses the x axis and ends at the point labeled A. Point A has coordinates negative 2 comma negative 3.

Think of the segment $AB$ as the hypotenuse of a right triangle. The legs can be drawn in as horizontal and vertical line segments.

A triangle is graphed in the coordinate plane with the origin labeled “O”. On the x-axis, the numbers negative 8 through negative one are indicated. On the y-axis, the numbers negative 3 through 4 are indicated. Two of the vertices, point A and point B, of the triangle are labeled. Point A is located at negative 2 comme negative 3 and point B is located at negative 8 comma 4. A vertical line is drawn from Point B directly down and a horizontal line is drawn from Point A to the left until the two lines meet creating the third vertex of the triangle. The two lines meet at the point with coordinates negative 8 comma negative 3. The vertical line is labeled with the text "the absolute value of four minus negative three equals 7". The horizontal line is labeled with the text "the absolute value of -8 minus -2 equals 6."

This length is a little longer than 9, since 85 is a little longer than 81. Using a calculator gives a more precise answer, $\sqrt{85} \approx 9.22$.

For a square, its side length is the square root of its area. For example, this square has an area of 16 square units and a side length of 4 units.

Both of these equations are true:

$\displaystyle 4^2=16$

$\displaystyle \sqrt{16}=4$

For a cube, the edge length is the cube root of its volume. For example, this cube has a volume of 64 cubic units and an edge length of 4 units:

Both of these equations are true:

$\displaystyle 4^3=64$

$\displaystyle \sqrt[3]{64}=4$

$\sqrt[3]{64}$ is pronounced “the cube root of 64.” Here are some other values of cube roots:

$\sqrt[3]{8}=2$ because $2^3=8$

$\sqrt[3]{27}=3$ because $3^3=27$

$\sqrt[3]{125}=5$ because $5^3=125$