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 
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
$$
√{(x_2x_1)^2+(y_2y_1)^2}
$$
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.
 $\cl"red"a$  $\cl"red"b$  $\cl"red"c$  $\cl"blue"d$
