Name __________________________
L ea rn i n g Obj e cti v e
Today, we will use the Pythagorean Theorem1 to find the length of
the missing side of a right triangle.
1
statement which can be proven mathematically
CFU
What are we going to do today?
What is a theorem?
Acti va t e ( o r p r o vi de ) P ri o r K n o wl edg e
A right triangle is a triangle with the largest angle equal to 90°.
• The hypotenuse is the side opposite the right angle.
• The legs are the two shorter sides of the right triangle.
(leg)
(hypotenuse)
(leg)
x  x2
13 169
14 196
Use the chart to evaluate the following.
1. 18
2
2
2. 15
15 225
16 256
17 289
18 324
3.
361
4.
196
19 361
20 400
Full chart is
located in the
back of the
lesson.
CFU
On your whiteboards, draw a right triangle. Label the hypotenuse. Label the legs. How
do you remember which side is the hypotenuse? How do you remember which sides
are the legs? Today, we will find the lengths of the hypotenuse and legs of right
triangles by using the Pythagorean Theorem.
DataWORKS Educational Research
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©2011 All rights reserved.
Comments? [email protected]
Geometry 15.0 (2Q)
Students use the Pythagorean theorem to determine distances and find missing
lengths of sides of right triangles.
Lesson to be used by EDI-trained teachers only.
Co n c e pt D ev el opm e n t
The Pythagorean Theorem is a mathematical formula2 for finding the side lengths of a
right triangle.
2
rule written in numbers and letters
● The Pythagorean Theorem states that the sum of the square
of legs a and b is equal to the square of the hypotenuse, c.
a b c
2
2
c
(hypotenuse)
a
(leg)
2
b (leg)
● The Pythagorean Theorem is commonly used to find the
length of a missing side of a right triangle.
Example:
Non-Example:
3 in
32  42  52
4 in
No right (90˚) angle
a2  b2  c 2
5 in
6 cm
9  16  25
25  25
8 cm
CFU
For which triangle could you use the Pythagorean Theorem to find the length of the missing side C?
Explain.
A.
B.
3 cm
3 cm
C
C
6 cm
4 cm
In your own words, what is the Pythagorean Theorem? The Pythagorean Theorem is ________________.
DataWORKS Educational Research
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©2011 All rights reserved.
Comments? [email protected]
Geometry 15.0 (2Q)
Students use the Pythagorean theorem to determine distances and find missing
lengths of sides of right triangles.
Lesson to be used by EDI-trained teachers only.
Imp o rta n c e
The Pythagorean Theorem is a mathematical formula for finding the side lengths of a
right triangle.
It is important to use the Pythagorean Theorem to find the length of
the missing side of a right triangle because:
1. using the Pythagorean Theorem will help you find the distance
between locations.
c ?
a
How long is the Salinas River
from King City to Soledad?
b
2. using the Pythagorean Theorem will help you do well on tests.
CFU
Does anyone else have another reason why it is important to use the Pythagorean Theorem to find the
length of the missing side of a right triangle? (pair-share) Why is it important to use the Pythagorean
Theorem to find the length of the missing side of a right triangle? You may give me one of my reasons or
one of your own. Which reason means more to you? Why?
DataWORKS Educational Research
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©2011 All rights reserved.
Comments? [email protected]
Geometry 15.0 (2Q)
Students use the Pythagorean theorem to determine distances and find missing
lengths of sides of right triangles.
Lesson to be used by EDI-trained teachers only.
Ski l l Dev el o p m en t /G u i ded P ra cti c e
The Pythagorean Theorem is a mathematical formula for
finding the side lengths of a right triangle.
● The Pythagorean Theorem states that the
a2  b2  c 2
sum of the square of legs a and b is equal
to the square of the hypotenuse, c.
a
(leg)
c
(hypotenuse)
b (leg)
Use the Pythagorean Theorem to find the length of the missing side of a right triangle.
Step #1: Identify the right angle.
Step #2: Determine which angle is the hypotenuse. (Hint: The hypotenuse is across from the right angle.)
Step #3: Find the length of the unknown side by using the Pythagorean Theorem.
a. Substitute3 the two known values into the Pythagorean Theorem formula.
Step #4: Solve4 the formula.
3
replace
4
find the answer
1. Find the length of hypotenuse YZ . ( YZ
means the line segment starting at point Y and ending at point Z.)
Z
12 cm
Y
9 cm
X
2. Find the length of hypotenuse XZ .
Z
5 in
X
13 in
12
Y
CFU How did I know which angle was the right angle? How did I identify the hypotenuse? How did I
know where to substitute the values? How did I solve the formula? How did you identify the right angle?
How did you identify the hypotenuse? How did you know where to substitute the values? How did you
solve the formula?
DataWORKS Educational Research
(800) 495-1550 • www.dataworks-ed.com
©2011 All rights reserved.
Comments? [email protected]
Geometry 15.0 (2Q)
Students use the Pythagorean theorem to determine distances and find missing
lengths of sides of right triangles.
Lesson to be used by EDI-trained teachers only.
Ski l l Dev el o pm en t /G u i ded P ra cti c e ( c on t i n u ed)
The Pythagorean Theorem is a mathematical formula for
finding the side lengths of a right triangle.
● The Pythagorean Theorem states that the
a2  b2  c 2
sum of the square of legs a and b is equal
to the square of the hypotenuse, c.
a
(leg)
c
(hypotenuse)
b (leg)
Use the Pythagorean Theorem to find the length of the missing side of a right triangle.
Step #1: Identify the right angle.
Step #2: Determine which angle is the hypotenuse. (Hint: The hypotenuse is across from the right angle.)
Step #3: Find the length of the unknown side by using the Pythagorean Theorem.
a. Substitute the two known values into the Pythagorean Theorem formula.
Step #4: Solve the formula.
3. Find the length of leg b.
10 in
6 in
b
4. Find the length of leg b.
5 cm
4 cm
b
CFU How did I know which angle was the right angle? How did I identify the hypotenuse? How did I
know where to substitute the values? How did I solve the formula? How did you identify the right angle?
How did you identify the hypotenuse? How did you know where to substitute the values? How did you
solve the formula?
DataWORKS Educational Research
(800) 495-1550 • www.dataworks-ed.com
©2011 All rights reserved.
Comments? [email protected]
Geometry 15.0 (2Q)
Students use the Pythagorean theorem to determine distances and find missing
lengths of sides of right triangles.
Lesson to be used by EDI-trained teachers only.
Ski l l Dev el o pm en t /G u i ded P ra cti c e ( c on t i n u ed)
The Pythagorean Theorem is a mathematical formula for
finding the side lengths of a right triangle.
● The Pythagorean Theorem states that the
a2  b2  c 2
sum of the square of legs a and b is equal
to the square of the hypotenuse, c.
c
(hypotenuse)
a
(leg)
b (leg)
Use the Pythagorean Theorem to find the length of the missing side of a right triangle.
Step #1: Identify the right angle.
Step #2: Determine which angle is the hypotenuse. (Hint: The hypotenuse is across from the right angle.)
Step #3: Find the length of the unknown side by using the Pythagorean Theorem.
a. Substitute the two known values into the Pythagorean Theorem formula.
Step #4: Solve the formula.
B
5. In the figure below, D is the midpoint of AC ,
and BD is perpendicular to AC . What is the length of BD ?
26 in
A
D
C
20 in
F
6. In the figure below, H is the midpoint of EG ,
and FH is perpendicular to EG . What is the length of FH ?
20 in
E
H
G
24 in
CFU How did I know which angle was the right angle? How did I identify the hypotenuse? How did I
modify the given information to find the length of the leg? How did I know where to substitute the values?
How did I solve the formula? How did you know which angle was the right angle? How did you identify the
hypotenuse? How did you modify the given information to find the length of the leg. How did you know
where to substitute the values? How did you solve the formula?
DataWORKS Educational Research
(800) 495-1550 • www.dataworks-ed.com
©2011 All rights reserved.
Comments? [email protected]
Geometry 15.0 (2Q)
Students use the Pythagorean theorem to determine distances and find missing
lengths of sides of right triangles.
Lesson to be used by EDI-trained teachers only.
Ski l l Dev el o pm en t /G u i ded P ra cti c e ( c on t i n u ed)
The Pythagorean Theorem is a mathematical formula for
finding the side lengths of a right triangle.
● The Pythagorean Theorem states that the
a2  b2  c 2
sum of the square of legs a and b is equal
to the square of the hypotenuse, c.
a
(leg)
c
(hypotenuse)
b (leg)
Use the Pythagorean Theorem to find the length of the missing side of a right triangle.
Step #1: Identify the right angle.
Step #2: Determine which angle is the hypotenuse. (Hint: The hypotenuse is across from the right angle.)
Step #3: Find the length of the unknown side by using the Pythagorean Theorem.
a. Substitute the two known values into the Pythagorean Theorem formula.
Step #4: Solve the formula.
7. In the figure below, M is the midpoint of JL ,
and KM is perpendicular to JL . KM is 15 cm.
JL is 16 cm. What is the length of JK ?
J
M
K
L
8. In the figure below, Q is the midpoint of OP ,
and NQ is perpendicular to OP . NQ is 24 cm.
O
OP is 14 cm. What is the length of ON ?
Q
N
P
CFU How did I know which angle was the right angle? How did I identify the hypotenuse? How did I
modify the given information to find the length of the leg? How did I know where to substitute the values?
How did I solve the formula? How did you know which angle was the right angle? How did you identify the
hypotenuse? How did you modify the given information to find the length of the leg. How did you know
where to substitute the values? How did you solve the formula?
DataWORKS Educational Research
(800) 495-1550 • www.dataworks-ed.com
©2011 All rights reserved.
Comments? [email protected]
Geometry 15.0 (2Q)
Students use the Pythagorean theorem to determine distances and find missing
lengths of sides of right triangles.
Lesson to be used by EDI-trained teachers only.
Cl osu r e
1. In your own words, what is the Pythagorean Theorem?
2. Use the Pythagorean Theorem to find the length of the missing side of the right triangle below.
3. What did you learn today about using the Pythagorean Theorem to find the length of the missing
side of a right triangle? Why is that important to you? (pair-share)
Step #1: Identify the right angle.
Step #2: Determine which angle is the hypotenuse. (Hint: The hypotenuse is across from the right angle.)
Step #3: Find the length of the unknown side by using the Pythagorean Theorem.
a. Substitute the two known values into the Pythagorean Theorem formula.
Step #4: Solve the formula.
a b c
2
2
2
a
(leg)
c
(hypotenuse)
b (leg)
1. Find the length of hypotenuse c.
c
8 cm
DataWORKS Educational Research
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©2011 All rights reserved.
Comments? [email protected]
2. Find the length of leg b.
1525inin
6 cm
20
12inin
b
Geometry 15.0 (2Q)
Students use the Pythagorean theorem to determine distances and find missing
lengths of sides of right triangles.
Lesson to be used by EDI-trained teachers only.
Name __________________________
In d ep en d en t P ra cti c e
The Pythagorean Theorem is a mathematical formula for
finding the side lengths of a right triangle.
● The Pythagorean Theorem states that the
a2  b2  c 2
sum of the square of legs a and b is equal
to the square of the hypotenuse, c.
a
(leg)
c
(hypotenuse)
b (leg)
Use the Pythagorean Theorem to find the length of the missing side of a right triangle.
Step #1: Identify the right angle.
Step #2: Determine which angle is the hypotenuse. (Hint: The hypotenuse is across from the right angle.)
Step #3: Find the length of the unknown side by using the Pythagorean Theorem.
a. Substitute the two known values into the Pythagorean Theorem formula.
Step #4: Solve the formula.
1. Find the length of hypotenuse ST .
S
8 ft
R
15 ft
T
2. Find the length of leg b.
25 in
20 in
b
DataWORKS Educational Research
(800) 495-1550 • www.dataworks-ed.com
©2011 All rights reserved.
Comments? [email protected]
Geometry 15.0 (2Q)
Students use the Pythagorean theorem to determine distances and find missing
lengths of sides of right triangles.
Lesson to be used by EDI-trained teachers only.
In d ep en d en t P ra cti c e ( co n ti n u ed)
The Pythagorean Theorem is a mathematical formula for
finding the side lengths of a right triangle.
● The Pythagorean Theorem states that the
a2  b2  c 2
sum of the square of legs a and b is equal
to the square of the hypotenuse, c.
a
(leg)
c
(hypotenuse)
b (leg)
Use the Pythagorean Theorem to find the length of the missing side of a right triangle.
Step #1: Identify the right angle.
Step #2: Determine which angle is the hypotenuse. (Hint: The hypotenuse is across from the right angle.)
Step #3: Find the length of the unknown side by using the Pythagorean Theorem.
a. Substitute the two known values into the Pythagorean Theorem formula.
Step #4: Solve the formula.
3. In the figure below, W is the midpoint of XZ ,
and YW is perpendicular to XZ . What is the length of YW ?
Y
15 in
X
Z
W
24 in
4. In the figure below, B is the midpoint of AC ,
and BD is perpendicular to AC . BD is 4 cm.
AC is 6 cm. What is the length of AD ?
A
B
D
C
DataWORKS Educational Research
(800) 495-1550 • www.dataworks-ed.com
©2011 All rights reserved.
Comments? [email protected]
Geometry 15.0 (2Q)
Students use the Pythagorean theorem to determine distances and find missing
lengths of sides of right triangles.
Lesson to be used by EDI-trained teachers only.
Name __________________________
P eri odi c R evi ew 1
The Pythagorean Theorem is a mathematical formula for
finding the side lengths of a right triangle.
● The Pythagorean Theorem states that the
a2  b2  c 2
sum of the square of legs a and b is equal
to the square of the hypotenuse, c.
a
(leg)
c
(hypotenuse)
b (leg)
Use the Pythagorean Theorem to find the length of the missing side of a right triangle.
Step #1: Identify the right angle.
Step #2: Determine which angle is the hypotenuse. (Hint: The hypotenuse is across from the right angle.)
Step #3: Find the length of the unknown side by using the Pythagorean Theorem.
a. Substitute the two known values into the Pythagorean Theorem formula.
Step #4: Solve the formula.
1. Find the length of hypotenuse c. (m = meters)
10 m
c
24 m
2. Find the length of leg a.
25 in
a
24 in
DataWORKS Educational Research
(800) 495-1550 • www.dataworks-ed.com
©2011 All rights reserved.
Comments? [email protected]
Geometry 15.0 (2Q)
Students use the Pythagorean theorem to determine distances and find missing
lengths of sides of right triangles.
Lesson to be used by EDI-trained teachers only.
P eri odi c R evi ew 1 ( c on ti n u e d)
The Pythagorean Theorem is a mathematical formula for
finding the side lengths of a right triangle.
● The Pythagorean Theorem states that the
a2  b2  c 2
sum of the square of legs a and b is equal
to the square of the hypotenuse, c.
a
(leg)
c
(hypotenuse)
b (leg)
Use the Pythagorean Theorem to find the length of the missing side of a right triangle.
Step #1: Identify the right angle.
Step #2: Determine which angle is the hypotenuse. (Hint: The hypotenuse is across from the right angle.)
Step #3: Find the length of the unknown side by using the Pythagorean Theorem.
a. Substitute the two known values into the Pythagorean Theorem formula.
Step #4: Solve the formula.
3. In the figure below, O is the midpoint of PQ ,
and NO is perpendicular to PQ . NO is 20 cm.
PQ is 30 cm. What is the length of NP ?
P
O
N
Q
4. In the figure below, M is the midpoint of LK ,
and JM is perpendicular to LK . What is the length of JM ?
J
17 m
L
M
K
30 m
DataWORKS Educational Research
(800) 495-1550 • www.dataworks-ed.com
©2011 All rights reserved.
Comments? [email protected]
Geometry 15.0 (2Q)
Students use the Pythagorean theorem to determine distances and find missing
lengths of sides of right triangles.
Lesson to be used by EDI-trained teachers only.
Name __________________________
P eri odi c R evi ew 2
The Pythagorean Theorem is a mathematical formula for
finding the side lengths of a right triangle.
● The Pythagorean Theorem states that the
a2  b2  c 2
sum of the square of legs a and b is equal
to the square of the hypotenuse, c.
a
(leg)
c
(hypotenuse)
b (leg)
Use the Pythagorean Theorem to find the length of the missing side of a right triangle.
Step #1: Identify the right angle.
Step #2: Determine which angle is the hypotenuse. (Hint: The hypotenuse is across from the right angle.)
Step #3: Find the length of the unknown side by using the Pythagorean Theorem.
a. Substitute the two known values into the Pythagorean Theorem formula.
Step #4: Solve the formula.
1. In the figure below, Y is the midpoint of XZ , and YW is perpendicular to XZ .
YW is 16 cm. XZ is 60 cm. What is the length of XW ?
X
Y
Z
W
2. Find the length of hypotenuse MN .
M
20 in
O
21 in
N
DataWORKS Educational Research
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©2011 All rights reserved.
Comments? [email protected]
Geometry 15.0 (2Q)
Students use the Pythagorean theorem to determine distances and find missing
lengths of sides of right triangles.
Lesson to be used by EDI-trained teachers only.
P eri odi c R evi ew 2 ( c on ti n u e d)
The Pythagorean Theorem is a mathematical formula for
finding the side lengths of a right triangle.
● The Pythagorean Theorem states that the
a2  b2  c 2
sum of the square of legs a and b is equal
to the square of the hypotenuse, c.
c
(hypotenuse)
a
(leg)
b (leg)
Use the Pythagorean Theorem to find the length of the missing side of a right triangle.
Step #1: Identify the right angle.
Step #2: Determine which angle is the hypotenuse. (Hint: The hypotenuse is across from the right angle.)
Step #3: Find the length of the unknown side by using the Pythagorean Theorem.
a. Substitute the two known values into the Pythagorean Theorem formula.
Step #4: Solve the formula.
3. In the figure below, D is the midpoint of CB ,
and AD is perpendicular to CB . What is the length of AD ?
A
5 in
C
D
B
6 in
4. Find the length of leg b.
b
18 ft
30 ft
DataWORKS Educational Research
(800) 495-1550 • www.dataworks-ed.com
©2011 All rights reserved.
Comments? [email protected]
Geometry 15.0 (2Q)
Students use the Pythagorean theorem to determine distances and find missing
lengths of sides of right triangles.
Lesson to be used by EDI-trained teachers only.
Name __________________________
P eri odi c R evi ew 3
The Pythagorean Theorem is a mathematical formula for
finding the side lengths of a right triangle.
● The Pythagorean Theorem states that the
a2  b2  c 2
sum of the square of legs a and b is equal
to the square of the hypotenuse, c.
a
(leg)
c
(hypotenuse)
b (leg)
Use the Pythagorean Theorem to find the length of the missing side of a right triangle.
Step #1: Identify the right angle.
Step #2: Determine which angle is the hypotenuse. (Hint: The hypotenuse is across from the right angle.)
Step #3: Find the length of the unknown side by using the Pythagorean Theorem.
a. Substitute the two known values into the Pythagorean Theorem formula.
Step #4: Solve the formula.
1. In the figure below, K is the midpoint of JL , and KM is perpendicular to JL .
KM is 8 cm. JL is 12 cm. What is the length of JM ?
J
K
L
M
2. Find the length of leg a.
36 cm
a
39 cm
DataWORKS Educational Research
(800) 495-1550 • www.dataworks-ed.com
©2011 All rights reserved.
Comments? [email protected]
Geometry 15.0 (2Q)
Students use the Pythagorean theorem to determine distances and find missing
lengths of sides of right triangles.
Lesson to be used by EDI-trained teachers only.
P eri odi c R evi ew 3 ( c on ti n u e d)
The Pythagorean Theorem is a mathematical formula for
finding the side lengths of a right triangle.
● The Pythagorean Theorem states that the
a2  b2  c 2
sum of the square of legs a and b is equal
to the square of the hypotenuse, c.
c
(hypotenuse)
a
(leg)
b (leg)
Use the Pythagorean Theorem to find the length of the missing side of a right triangle.
Step #1: Identify the right angle.
Step #2: Determine which angle is the hypotenuse. (Hint: The hypotenuse is across from the right angle.)
Step #3: Find the length of the unknown side by using the Pythagorean Theorem.
a. Substitute the two known values into the Pythagorean Theorem formula.
Step #4: Solve the formula.
3. Find the length of hypotenuse c.
24 m
18 m
c
4. In the figure below, E is the midpoint of DF ,
and GE is perpendicular to DF . What is the length of GE ?
G
15 in
D
E
F
18 in
DataWORKS Educational Research
(800) 495-1550 • www.dataworks-ed.com
©2011 All rights reserved.
Comments? [email protected]
Geometry 15.0 (2Q)
Students use the Pythagorean theorem to determine distances and find missing
lengths of sides of right triangles.
Lesson to be used by EDI-trained teachers only.
P eri odi c R evi ew
x  x2
x  x2
x  x2
1
1
11 121
21 441
2
4
12 144
22 484
3
9
13 169
23 529
4
16
14 196
5
25
6
x  x2
31
1
32
961
1024
1089
24 576
33
3
34
15 225
25 625
35
1225
36
16 256
26 676
36
1296
7
49
17 289
27 729
37
1369
8
64
18 324
28 784
38
1444
9
81
19 361
29 841
39
1521
10 100
20 400
30 900
40
1600
1156
Example.
The square root of 361 is 19.
361  19
19 squared is 361.
19 2  361

DataWORKS Educational Research
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©2011 All rights reserved.
Comments? [email protected]
Geometry 15.0 (2Q)
Students use the Pythagorean theorem to determine distances and find missing
lengths of sides of right triangles.
Lesson to be used by EDI-trained teachers only.
Co n c e pt D em on st ra t i on
The Pythagorean Theorem states
that the sum of the square of legs a and b is equal to the square of the hypotenuse, c.
3
5
4
2
a
2
+ b = c
DataWORKS Educational Research
(800) 495-1550 • www.dataworks-ed.com
©2011 All rights reserved.
Comments? [email protected]
2
Geometry 15.0 (2Q)
Students use the Pythagorean theorem to determine distances and find missing
lengths of sides of right triangles.
Lesson to be used by EDI-trained teachers only.