Welcome to a lesson on the Pythagorean Theorem. The mathematician Pythagoras
prove the Pythagorean theorem. The theorem states that given any right
triangle besides a, b, and c as below, the following relationship is always true
about the lengths of the sides: a-squared plus b-squared equals c-squared.
The triangle must be a right triangle, which means it contains a ninety degree angle.
The side c is called the hypotenuse and sits across from the right angle.
The hypotenuse is also the longest side of the triangle
It's important that we identify the hypotenuse correctly because looking at
the equation, notice how it's on one side of the equation by itself.
The lengths a and b form the right angle and are interchangeable in the theorem.
Again notice how the sides a and b form the right angle.
Let's look at some examples.
We're asked to use the Pythagorean Theorem to find the length of the missing side and
round to the tenths place value. So notice right away,
the side opposite the right angle is the hypotenuse and it's already labeled c correctly.
So now we can label the two legs, a and b, it does not matter which leg
is a and which leg is b.
So let's call this leg a and this leg b. And now we'll use
the Pythagorean Theorem to determine the length of the hypotenuse. The Pythagorean
Theorem is a squared plus b-squared equals c-squared.
So we substitute seven for a and twelve for b.
this gives us seven squared plus twelve squared
equals c squared. And now we simplify the left side: seven squared equals seven times seven,
which equals forty-nine. Twelve squared equals twelve times twelve, which equals one hundred forty-four.
Now we find the sum: forty-nine plus one hundred forty-four equals one hundred ninety-three.
We have one hundred ninety-three equals c-squared.
Our goal is to find
c, not c-squared. Because c-squared is a perfect square, to solve for c, we take the
principal square root of both sides of the equation. We know c has to be
positive because it's a length and therefore, this is the principal square root.
The square root of c squared is equal to one factor of c, so c is equal to the
square root of one hundred ninety-three.
One hundred ninety-three is not a perfect square and therefore, we need to use the
calculator to get a decimal approximation for c.
Going to the calculator,
we enter the square root,
two hundred ninety-three, enter.
To the tenths place value notice how we round up. Because of the nine in the
hundreds place value,
this is approximately thirteen point nine, rounded to the tenths place value .
Which meas the length of the hypotenuse in the right triangle
is approximately thirteen point nine.
Normally we do include the units, but in this case we were
not given units.
For b, we're asked to sketch the right triangle and find the length of the side not given.
Notice how we're given the lengths of the two legs.
So let's go ahead and sketch a right triangle, the two legs of the sides that
form the right angle.
Let's label the short leg eight centimeters and the long leg twelve
centimeters.
Notice the missing length is the hypotenuse, which must be labeled c.
Let's label the legs a and b, which can be interchanged. Let's call the short
leg a and long leg b.
And now we'll perform substitution into the Pythagorean Theorem to solve for c.
A-squared is eight squared,
plus b-squared is twelve squared,
and this must equals c-squared.
Eight squared equals sixty-four.
Twelve squared equals one hundred forty-four.
Sixty-four plus one hundred forty-four is two hundrd eight. Two hundred eight equals c-squared. To solve for c, we take the principal
square root of both sides of the equation. Again because we know c must
be positive,
the square root of c squared equals one factor of c. Two hundred eight is not a
perfect square and therefore, we will get a decimal approximation using the
calculator. We are asked to round to the nearest hundredth,
which means two decimal places.
So we enter the square root
of two hundred eight,
enter.
Notice there's a two in the third decimal place, so we round down
this is approximately fourteen point two rounded to the hundreds place value.
So because c is approximately fourteen point four, two, the length of the hypotenuse
is approximately fourteen point four, two centimeters.
I hope you found this helpful.