Download 2.5

Section 2.5 – Right Triangles
Right triangles have some special properties that other types of triangles do not. One of those is called the
Pythagorean Theorem. Before we examine them more closely, we need to look at square roots.
§1 Square Roots
The square root of a number is the number we must square to get that number. It is usually given by the square
root sign
. The following is a list of the first twelve square roots.
1 1=
49 7=
16 4 =
100 10
=
4 2=
64 8=
25 5 =
121 11
=
9 3=
81 9 =
36 6 =
144 12
=
These are called perfect squares. For any other number, we can use a calculator and round to the nearest
hundredth.
§2 Pythagorean Theorem
In a right triangle, the longest side is called the hypotenuse. The other two sides are called legs.
Usually, the hypotenuse is denoted by the letter, c, while the legs are called a and b.
2
The formula is then c=
a 2 + b2
For example, if the legs of a right triangle are 3 inches and 4 inches, how long is the hypotenuse?
2
Let a = 3 and let b = 4. Plug these values into the formula to get c=
32 + 42 . We end up with c = 5 .
NOTE: Sometimes the answer may not be a perfect square. You may have to use a calculator.
For example, if the legs of a right triangle are 4 feet and 6 feet, how long is the hypotenuse?
2
Let a = 4 and let b = 6. Plug these into the formula to get c=
42 + 62 , hence c 2 =16 + 36 = 52 . Note that if
c 2 = 52, then c = 52 . Use your calculator to find the answer, 7.21.
PRACTICE
1) If c 2 = 49 , what does c equal?
2) If c 2 = 71 , what does c equal?
3) In a right triangle, if a = 6 and b = 8, what is the length of the hypotenuse?
4) In a right triangle, if a = 10 and b = 10, what is the length of the hypotenuse?
5) In a right triangle, if the hypotenuse is 13 inches and one leg is 12 inches, what is the length of the other leg?
§3 Some applications of the Pythagorean Theorem
There are some numbers that work out ‘nice’ in the Pythagorean Theorem. We call these Pythagorean triples,
because there are three numbers. When we say the numbers work out nice, we mean that none of the numbers
are decimals. For example, we’ve seen that when a = 3 and b = 4, then c =5. This is one example of a
Pythagorean triple. The following is a list of some of the more common triples:
3− 4−5
5 − 12 − 13
7 − 24 − 25
8 − 15 − 17
One important note is, any multiple of these is also a triple. So we can multiply any of the triples by a nonnegative number and they will still work !
For example, what if a = 6, b= 8, and c = 10. Does 62 + 82 =
102 ? What about 92 + 122 =
152 ?
PRACTICE
6) If you drive 5 miles due east, then make a left turn and drive 12 miles due north, how far are you in a straight
line from your starting point?
7) A television screen is 40 inches, meaning the diagonal length is 40 inches. If the length of the TV is 23 inches,
what is the height of the TV?