Section 3-6
Pythagorean Theorem Notes
In a right
triangle the two sides that form the right angle are called the legs of the
triangle and the side opposite the right angle is called the hypotenuse.
In our graphic AB and BC are
the two legs and AC is the hypotenuse.
In all right triangles the hypotenuse will be
the longest side.
Let us say that the length of AB is a,
the length of BC is b, and the length of AC is c
The Pythagorean Theorem states
|
In a right triangle the
sum of the square of the lengths of the legs is equal to the square of the
hypotenuse. a 2 + b
2 = c 2 |
If the length of one leg a = 3 and the
length of the other leg b = 4 what will be the length of c the hypotenuse?
|
a 2 + b
2 = c 2 |
1. Write the formula |
|
3 2 + 4
2 = c 2 |
2.
Substitute the given lengths in place of the variables |
|
9 + 16
= c 2 |
3.
Calculate the square of each leg |
|
25 =
c 2 |
4.
Add the squares of the legs |
|
|
5.
Take the square root of each side |
|
5 = c |
6.
The length of the hypotenuse is 5 |
If the length of one leg a = 3 and the length
of c the hypotenuse = 5 what will be the length of the other leg b?
|
a 2 + b
2 = c 2 |
1. Write the formula |
|
3 2 + b
2 = 5 2 |
2. Substitute the given
lengths in place of the variables |
|
9 + b 2 = 25 |
3. Calculate the square of
each given measurement |
|
9 - 9 + b 2 = 25
– 9 |
4. Isolate the variable |
|
b 2 = 16 |
|
|
|
5. Take the square root of
each side |
|
b = 4 |
6. The length of the leg b is
4 |
The converse of the Pythagorean Theorem
|
If a, b and c are the
lengths of the sides of a triangle and a 2 + b 2 = c
2 then the
triangle is a right triangle. |
Can the given lengths 7, 25, 24 be the lengths
of the sides of a right triangle?
Since
we know that the hypotenuse is always the longest side I know that they
hypotenuse (c in our formula) = 25.
It
does not matter which of the other two lengths I substitute for a
or b
|
a 2 + b
2 = c 2 |
1. Write the formula |
|
7 2 + 24
2 = 25 2 |
2. Substitute the given
lengths in place of the variables |
|
49 + 576
= 625 |
3. Calculate the square of
each given measurement |
|
625 = 625 |
4. Calculate the sum of the squares of the legs
|
|
If the two figures are
equal then the triangle is a right triangle! Yes we have a right triangle with the legs
measuring 7 and 24 and with a hypotenuse of 25. |
|
Can the given lengths 12, 8, 9 be the lengths
of the sides of a right triangle?
Since
we know that the hypotenuse is always the longest side I know that they
hypotenuse (c in our formula) = 12.
It
does not matter which of the other two lengths I substitute for a
or b
|
a 2 + b
2 = c 2 |
1. Write the formula |
|
9 2 + 8
2 = 12 2 |
2. Substitute the given
lengths in place of the variables |
|
81 + 64
= 144 |
3. Calculate the square of
each given measurement |
|
145 = 144 |
4. Calculate the sum of the squares of the legs
|
|
If the two figures are
equal then the triangle is a right triangle! No we do not have a right triangle because 9 2 + 8
2 does not equal 12 2 |
|