geometry 9.3 converse of the pythagorean theorem
Post on 12-Jan-2016
249 Views
Preview:
TRANSCRIPT
Geometry
9.3 Converse of the Pythagorean Theorem
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 2
Goals
I can determine if a triangle is a right triangle.
I can use the Pythagorean inequalities to determine if a triangle is acute or obtuse.
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 3
Pythagorean Theorem In a right triangle, the square of
the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
If ABC is a right triangle, then a2 + b2 = c2
a
b
c
A
B
C
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 4
Converse of Pythagorean Theorem
If the square of the length of the longest side of a 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 a2 + b2 = c2, then ABC is a right triangle.
a
b
c
A
B
C
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 5
Example 1
Is POD a right triangle?
P
O
D
30 16
34
?2 2 2
?
16 30 34
256 900 1156
1156 1156
Yes!Longest Side
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 6
Reminder
22
2
2
5 5
17 17
3 3
x x
x x
2 2 2 2
2 22
2 22
2
3 3 9
3 3 9
3 3 3 3 9 3 27
4 5 16 5 80
x x x
x x x
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 7
Example 2
Is HUG a right triangle?
5 5
H
U G
5
10
Which segment is the longest? HG
? 22 2
? 22
?
5 10 5 5
25 100 5 5
125 25 5
125 125
Yes!
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 8
Example 3
Is SAD a right triangle?
S
A D
9
12
Which segment is the longest? SD
?2 2 2
?
9 12 20
81 144 400
225 400
No!
20
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 9
Your Turn.
Is RST a right ?
R
S
T26
1024
?2 2 2
?
10 24 26
100 576 676
676 676
Yes it is.
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 10
Triangle Inequality Theorem
In a triangle, the sum of any two sides is greater than the third side.
4
5
74 + 7 > 5
4 + 5 > 7
5 + 7 > 4
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 11
Triangle Inequality Theorem
5
10
4This is not a triangle since 5 + 4 < 10.
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 12
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 13
a
b
c
Begin with a right triangle…
a2 + b2 = c2
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 14
a
b
ca
c
a and b have not changed.
a2 + b2 has not changed.
c got smaller.
c2 got smaller.
and…
The right angle gets smaller: it is acute.
Rotate side a in.
c2 = a2 + b2c2 < a2 + b2
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 15
Theorem 9.6
If the square of the length of the longest side of a triangle is less than the sum of the squares of the other two sides, then the triangle is acute.
A
BC a
bc
c2 < a2 + b2
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 16
a
b
c
Take another right triangle…
a2 + b2 = c2
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 17
a
b
ca c
a and b have not changed.
a2 + b2 has not changed.
c got larger.
c2 got larger.
and…
The right angle gets larger: it is obtuse.
Rotate side a out.
c2 = a2 + b2c2 > a2 + b2
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 18
Theorem 9.6 If the square of the length of the
longest side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is obtuse.
c2 > a2 + b2
A
BC a
bc
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 19
Example 4
The sides of a triangle measure 5, 7, and 11. Classify it as acute, right, or obtuse.
Solution: The longest side is 11. 112 ? 52 + 72
121 ? 25 + 49 121 > 74 Obtuse
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 20
Example 5
The sides of a triangle are 17, 20, and 25. Classify the triangle.
Solution: 252 ? 172 + 202
625 ? 689 625 < 689 Acute
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 21
Example 6
Classify this triangle.
57
12
2 2 212 ____ 7 5
12____ 7 5
12 12
?
?
Right
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 22
Example 7
Classify this triangle.
It isn’t a triangle! 6 +8 < 16.
68
16
April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 23
Summary
If c2 = a2 + b2, RIGHT . If c2 < a2 + b2, ACUTE . If c2 > a2 + b2, OBTUSE . The last two can be very confusing;
don’t get them mixed up.
top related