# Distances Using the Pythagorean Theorem

In this lesson, you will see how to use the Pythagorean Theorem to find the distance between two different points or locations.

## Diagonal of a rectangular area

The Eagleville High School grounds are a rectangular plot of land which is 1000 feet long and 700 feet wide. Its students would like to figure out how far it is between two opposite corners of the high school grounds: that is, they would like to find the length of a diagonal of the grounds. The grounds are drawn to the left, with a diagonal of length \$\cl"blue"d\$ drawn in blue.

 Is \$\cl"blue"d\$ shorter than the shortest edge (shorter than 700 feet), between the lengths of the two edges (between 700 and 1000 feet), or longer than the longest edge (longer than 1000 feet)?
 Because the grounds are a rectangle, all of its corners are right angles. Is the triangle \$▵ABC\$ a right triangle?
 Which side of the right triangle \$▵ABC\$ is its hypotenuse? (Which two vertices are that side’s ends?)

Using the Pythagorean Theorem, find the length \$\cl"blue"d\$ of the hypotenuse of \$▵ABC\$. Round your answer to the nearest foot.

 The distance between any two points is the length of the line segment connecting them. What is the distance between the points \$\cl"red"A\$ and \$\cl"red"C\$ (that is, the length of the line segment \$\cl"blue"\ov{AC}\$)? Round your answer to the nearest foot. feet

## Distance between two coordinate points

 The two points \$\cl"red"{A=(4, 4)}\$ and \$\cl"red"{B=(-2, 4)}\$ are plotted on the grid to the left. What is the distance between \$\cl"red"A\$ and \$\cl"red"B\$ (the length of the segment \$\cl"red"\ov{AB}\$)?
 Click . The two points \$\cl"red"{B=(-2, 4)}\$ and \$\cl"red"{C=(-2, -4)}\$ are plotted on the grid to the left. What is the distance between \$\cl"red"B\$ and \$\cl"red"C\$ (the length of the segment \$\cl"red"\ov{BC}\$)?
 Click . Now all three points are drawn. Because \$\cl"red"\ov{AB}\$ is horizontal and \$\cl"red"\ov{BC}\$ is vertical, \$\cl"red"{∠B}\$ is a right angle. Using the Pythagorean Theorem, find the length of the hypotenuse of the right triangle \$\cl"red"{▵ABC}\$.
 What is the distance between \$\cl"red"A\$ and \$\cl"red"C\$?

For each row in the table below, a right triangle \$\cl"red"{▵ABC}\$ with horizontal and vertical legs is drawn on the grid to the left. Use the lengths of its legs to find the length of its hypotenuse \$\cl"red"\ov{AC}\$ (the distance from \$\cl"red"A\$ to \$\cl"red"C\$). Round the length of the hypotenuse to one decimal place.

length of
\$\cl"red"\ov{AB}\$
length of
\$\cl"red"\ov{BC}\$
length of
\$\cl"red"\ov{AC}\$

If you have any two points \$A=(x_1,y_1)\$ and \$C=(x_2,y_2)\$, you can always fit them into a right triangle in this way. The lengths of the legs of the right triangle will be the horizontal and vertical difference between the two points, so we can use the Pythagorean theorem to get a formula for the distance between \$A\$ and \$C\$:

If \$A=(x_1,y_1)\$ and \$C=(x_2,y_2)\$ are two points in the coordinate plane, then the distance between \$A\$ and \$C\$ is \$\$ √{(x_2-x_1)^2+(y_2-y_1)^2} \$\$

Using the formula from the last question, find the distance between each pair of points in the table below. Round your answer to one decimal place.

\$\cl"red"{(x_1,y_1)}\$\$\cl"red"{(x_2,y_2)}\$distance

## Diagonal of a box

A rectangular box is drawn to the left. It is 3 units wide, 4 units long, and 12 units tall. Its sides, top, and bottom are all rectangles. We would like to discover the length \$\cl"blue"d\$ of its long diagonal.

 Is \$\cl"blue"d\$ shorter than the shortest edge (shorter than 3), between the length of the shortest edge and the length of the longest edge (between 3 and 12), or longer than the longest edge (longer than 12)?

Click to draw a green triangle along the bottom of the box.

 Is the green triangle a right triangle?
 What are the lengths of the legs of the green triangle?
 Using the Pythagorean Theorem, find the length \$\cl"green"e\$ of the green triangle’s hypotenuse.

Click to draw a blue triangle through the middle of the box. One of the sides of this triangle is an edge of the box, and another is the hypotenuse of the green triangle.

 Is the blue triangle a right triangle?
 What are the lengths of the legs of the blue triangle?
 Using the Pythagorean Theorem, find the length of the blue triangle’s hypotenuse.
 What is the length \$\cl"blue"d\$ of the box’s long diagonal?

You can find the length of the diagonal of any box in this way. If the box is \$a\$ units wide, \$b\$ units long, and \$c\$ units tall, and it has a bottom diagonal of length \$e\$, then \$e^2=a^2+b^2\$. If it has a long diagonal of length \$d\$, then \$d^2=e^2+c^2=a^2+b^2+c^2\$. So:

If the edges of a rectangular box have lengths \$a\$, \$b\$, and \$c\$, then the length of its long diagonal is \$\$ √{a^2+b^2+c^2} \$\$

Each row of the table below gives the three side lengths \$\cl"red"a\$, \$\cl"red"b\$, and \$\cl"red"c\$ of a rectangular box. (You can click and drag to rotate each box.) Find the length \$\cl"blue"d\$ of the long diagonal of each row’s box. Round your answer to one decimal place.

\$\cl"red"a\$\$\cl"red"b\$\$\cl"red"c\$\$\cl"blue"d\$