Transcript Slide 1

The Pythagorean Theorem
and Its Converse

Use the Pythagorean Theorem

Use the Converse of the Pythagorean
Theorem
Theorem 7.4
In a right triangle, the
square of the length
of the hypotenuse is
equal to the sum of
the squares of the
legs.
a2 + b2 = c2
STATEMENTS
REASONS
1. Draw a ┴ from C to AB
1. Perpendicular Postulate
c = a
1. a
2.
e
2. Geometric Mean Theorem
and
c = b
b
f
3. ce = a2 and cf = b2
3. Cross Product Property
4. ce + cf = a2 + b2
4. Addition Property
5. c(e + f) = a2 + b2
5. Distributive Property
6. e + f = c
6. Segment Add. Postulate
7. c2 = a2 + b2
7. Substitution Property
LONGITUDE AND LATITUDE Carson City, Nevada, is
located at about 120 degrees longitude and 39
degrees latitude. NASA Ames is located about 122
degrees longitude and 37 degrees latitude. Use the
lines of longitude and latitude to find the degree
distance to the nearest tenth degree if you were to
travel directly from NASA Ames to Carson City.
The change in longitude between NASA Ames and Carson
City is
or 2 degrees. Let this distance be a.
The change in latitude is
Let this distance be b.
or 2 degrees latitude.
Use the Pythagorean Theorem to find the distance in
degrees from NASA Ames to Carson City, represented
by c.
Pythagorean Theorem
Simplify.
Add.
Take the square root of
each side.
Use a calculator.
Answer: The degree distance between NASA Ames
and Carson City is about 2.8 degrees.
Find d.
Pythagorean Theorem
Simplify.
Subtract 9 from each side.
Take the square root of
each side.
Use a calculator.
Answer:
Find x.
Answer:

Theorem 7.5
If the square of the
length of the longest side
of the triangle is equal to
the sum of the squares
of the lengths of the
other two sides, then the
triangle is a right
triangle.
If c2 = a2 + b2, then
∆ABC is a right triangle.
COORDINATE GEOMETRY
Verify that
is a right triangle.
Use the Distance Formula to determine the lengths of the
sides.
Subtract.
Simplify.
Subtract.
Simplify.
Subtract.
Simplify.
By the converse of the Pythagorean Theorem, if the sum
of the squares of the measures of two sides of a triangle
equals the square of the measure of the longest side, then
the triangle is a right triangle.
Converse of the
Pythagorean Theorem
Simplify.
Add.
Answer: Since the sum of the squares of two sides
equals the square of the longest side,
is a right triangle.
COORDINATE GEOMETRY Verify that
triangle.
is a right
Answer:
is a
right triangle because

A Pythagorean Triple is three whole
numbers that satisfy the equation a2 + b2
= c2 , where the longest side is the
hypotenuse, c, and
a and b are the two legs.
i.e. 3, 4, 5
Determine whether 9, 12, and 15 are the sides of a
right triangle. Then state whether they form a
Pythagorean triple.
Since the measure of the longest side is 15, 15 must be c.
Let a and b be 9 and 12.
Pythagorean Theorem
Simplify.
Add.
Answer: These segments form the sides of a right
triangle since they satisfy the Pythagorean Theorem. The
measures are whole numbers and form a Pythagorean
triple.
Determine whether 21, 42, and 54 are the sides of a
right triangle. Then state whether they form a
Pythagorean triple.
Pythagorean Theorem
Simplify.
Add.
Answer: Since
, segments with these
measures cannot form a right triangle. Therefore,
they do not form a Pythagorean triple.
Determine whether
4, and 8 are the sides of a
right triangle. Then state whether they form a
Pythagorean triple.
Pythagorean Theorem
Simplify.
Add.
Answer: Since 64 = 64, segments with these measures
form a right triangle. However,
is not a whole number.
Therefore, they do not form a Pythagorean triple.
Determine whether each set of measures are the sides
of a right triangle. Then state whether they form a
Pythagorean triple.
a. 6, 8, 10
Answer: The segments form the sides of a right triangle
and the measures form a Pythagorean triple.
b. 5, 8, 9
Answer: The segments do not form the sides of a right
triangle, and the measures do not form a Pythagorean triple.
c.
Answer: The segments form the sides of a right triangle,
but the measures do not form a Pythagorean triple.