8.1 The Converse of the
Pythagorean Theorem
Geometry
Mr. Peebles
Spring 2013
Bell Ringer
You are to walk around the room
and locate and write the 10 BEs
for math success.
Bell Ringer
You are to walk around the room
and locate and write the 10 BEs
for math success.
Time’s Up!
Bell Ringer-Answer
The 10 Non-Negotiable “BEs” For Geometry Success
1.
2.
3.
4.
BE on time and in class every day.
BE dressing for success.
BE respectful… give it, earn it.
BE prepared with your own work and
materials.
5.
BE participating as a team player.
6.
BE meeting all course due dates.
7.
BE electronics, makeup, food, drink, candy, and gum-free.
8.
BE following all teacher directions.
9.
BE following all class, school, and district rules.
10.
BE expecting “necessary” changes.
Daily Learning Target (DLT)
• “I can apply my knowledge of right
triangles and Pythagorean Theorem to
determine if a triangle is a right triangle.”
Pre-Assessment
On the piece of paper, write down what you
Know about the following topics.
1.
2.
3.
4.
5.
6.
Converse of Pythagorean Theorem
45-45-90 Right Triangles
30-60-90 Right Triangles
Tangent Ratio
Sine Ratio
Cosine Ration
Using the Converse
• In the past, you learned that if a triangle is
a right triangle, then the square of the
length of the hypotenuse is equal to the
sum of the squares of the length of the
legs. The Converse of the Pythagorean
Theorem is also true, as stated on the
following slide.
Theorem 9.5: Converse of the
Pythagorean Theorem
• 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.
B
c
a
C
b
A
Note:
• You can use the Converse of the
Pythagorean Theorem to verify that a
given triangle is a right triangle, as shown
in Example 1.
Ex. 1: Verifying Right Triangles
• The triangles on the
slides that follow
appear to be right
triangles. Tell
whether they are right
triangles or not.
8
7
√113
4√95
15
36
Ex. 1a: Verifying Right Triangles
• Let c represent the
length of the longest
side of the triangle.
Check to see whether
the side lengths
satisfy the equation c2
= a 2 + b 2.
?
(√113)2 = 72 + 82
?
113 = 49 + 64
113 = 113 ✔
8
7
√113
The triangle is a
right triangle.
Ex. 1b: Verifying Right Triangles
c 2 = a 2 + b 2.
?
2
(4√95) = 152 + 362
?
42 ∙ (√95)2 = 152 + 362
?
16 ∙ 95 = 225+1296
1520 ≠ 1521 ✔
4√95
15
36
The triangle is NOT a
right triangle.
Classifying Triangles
• Sometimes it is hard to tell from looking at
a triangle whether it is obtuse or acute.
The theorems on the following slides can
help you tell.
Theorem 9.6—Triangle Inequality
• If the square of the
length of the longest
side of a triangle is
less than the sum of
the squares of the
lengths of the other
two sides, then the
triangle is acute.
• If c2 < a2 + b2, then
∆ABC is acute
A
c
b
C
a
c2 < a 2 + b2
B
Theorem 9.7—Triangle Inequality
• If the square of the
length of the longest
side of a triangle is
greater than the sum
of the squares of the
lengths of the other
two sides, then the
triangle is obtuse.
• If c2 > a2 + b2, then
∆ABC is obtuse
A
c
b
B
C
a
c2 > a 2 + b2
Ex. 2: Classifying Triangles
•
Decide whether the set of numbers can
represent the side lengths of a triangle. If they
can, classify the triangle as right, acute or
obtuse.
a. 38, 77, 86
b. 10.5, 36.5, 37.5
You can use the Triangle Inequality to confirm that
each set of numbers can represent the side
lengths of a triangle. Compare the square o
the length of the longest side with the sum of
the squares of the two shorter sides.
Triangle Inequality to confirm
Example 2a
Statement:
c 2 ? a2 + b2
862 ? 382 + 772
7396 ? 1444 + 5959
7395 > 7373
Reason:
Compare c2 with a2 + b2
Substitute values
Multiply
c2 is greater than a2 + b2
The triangle is obtuse
Triangle Inequality to confirm
Example 2b
Statement:
c 2 ? a2 + b2
37.52 ? 10.52 + 36.52
1406.25 ? 110.25 +
1332.25
1406.24 < 1442.5
Reason:
Compare c2 with a2 + b2
Substitute values
Multiply
c2 is less than a2 + b2
The triangle is acute
Ex. 3: Building a foundation
• Construction: You use
four stakes and string to
mark the foundation of a
house. You want to
make sure the foundation
is rectangular.
a. A friend measures the
four sides to be 30 feet,
30 feet, 72 feet, and 72
feet. He says these
measurements prove that
the foundation is
rectangular. Is he
correct?
Ex. 3: Building a foundation
• Solution: Your friend is not correct. The
foundation could be a nonrectangular
parallelogram, as shown below.
Ex. 3: Building a foundation
b. You measure one of the diagonals to be
78 feet. Explain how you can use this
measurement to tell whether the
foundation will be rectangular.
Ex. 3: Building a foundation
Solution: The diagonal
divides the foundation into
two triangles. Compare
the square of the length of
the longest side with the
sum of the squares of the
shorter sides of one of
these triangles.
• Because 302 + 722 =
782, you can conclude
that both the triangles are
right triangles. The
foundation is a
parallelogram with two
right angles, which
implies that it is
rectangular
Assignment
• Pgs. 420-423 (11-17 Odds, 21-29 All, 31,
54, 59, 60)
Closure:
• Given three sides, how would you know if
a triangle is acute, right, or obtuse?
- Please write your answer on the
whiteboards.
Closure:
• Given three sides, how would you know if
a triangle is acute, right, or obtuse?
- Please write your answer on the
whiteboards.
c2 < a2 + b2 = Acute Triangle
c2 = a2 + b2 = Right Triangle
c2 > a2 + b2 = Obtuse Triangle