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PYTHAGOREAN THEOREM WORD PROBLEMS
PYTHAGOREAN THEOREM WORD PROBLEMS
OBJECTIVE: Students will solve for missing values using the
Pythagorean theorem for word problems
OBJECTIVE: Students will solve for missing values using the
Pythagorean theorem for word problems
RECALL THE PYTHAGOREAN THEOREMRECALL THE PYTHAGOREAN THEOREM
� THE PYTHAGOREAN THEOREM STATES THAT FOR ANY RIGHT TRIANGLE:
a2 + b2 = c2
� THE PYTHAGOREAN THEOREM STATES THAT FOR ANY RIGHT TRIANGLE:
a2 + b2 = c2
Warm UpWarm Up� Solve for the missing side in each right
triangle below.� Remember: a2 + b2 = c2
� Solve for the missing side in each right triangle below.� Remember: a2 + b2 = c2
1.)2.)
c3
4
7
2
b
Warm Up ANSWERS Warm Up ANSWERS
� Solve for the missing side in each right triangle below.� Remember: a2 + b2 = c2
� Solve for the missing side in each right triangle below.� Remember: a2 + b2 = c2
1.2.
c3
4
7
2
b
32 + 42 = c2
25 = c2
c = 5
b2 + 22 = 72
b2 + 4 = 49b2 = 45b = √45 = 6.7
WORD PROBLEMSWORD PROBLEMS� Many Pythagorean Theorem Questions are given as
word problems.
STEPS FOR SOLVING PYTHAGOREAN THEOREM WORD PROBLEMS:
Step 1: Circle any important numbers in the problem
Step 2: Draw a picture of the situation.
Step 3: Solve using the Pythagorean theorem (a2 + b2 = c2)
� Many Pythagorean Theorem Questions are given as word problems.
STEPS FOR SOLVING PYTHAGOREAN THEOREM WORD PROBLEMS:
Step 1: Circle any important numbers in the problem
Step 2: Draw a picture of the situation.
Step 3: Solve using the Pythagorean theorem (a2 + b2 = c2)
EXAMPLE 1EXAMPLE 1
� Mr. Blonde is measuring his TV. The TV is 12 inches wide and 9 inches tall. What is the length of the diagonal on the TV?
� Mr. Blonde is measuring his TV. The TV is 12 inches wide and 9 inches tall. What is the length of the diagonal on the TV?
Step 1: Circle Important numbers.
Step 2: Draw a picture of the situation.
12 inches
9 in
ches
STEP 3: PYTHAGOREAN THEOREMa2 + b2 = c2
122 + 92 = c2
144 + 81 = c2
225 = c2
C = √ 225 = 15 in
EXAMPLE 2EXAMPLE 2
� Mr. Blue is measuring his TV. The TV is 12 inches wide and 5 inches tall. What is the length of the diagonal on the TV?
� Mr. Blue is measuring his TV. The TV is 12 inches wide and 5 inches tall. What is the length of the diagonal on the TV?
Step 1: Circle Important numbers.
Step 2: Draw a picture of the situation.
12 inches
5 in
ches
STEP 3: PYTHAGOREAN THEOREMa2 + b2 = c2
122 + 52 = c2
144 + 25 = c2
169 = c2
C = √ 169 = 13 in
EXAMPLE 3EXAMPLE 3
Tim & Sal have a 10 meter ladder. It must reach exactly 6 meters up a building, how far away from the building should the ladder be placed?
Tim & Sal have a 10 meter ladder. It must reach exactly 6 meters up a building, how far away from the building should the ladder be placed?
Step 1: Circle Important numbers.
Step 2: Draw a picture of the situation.STEP 3: PYTHAGOREAN THEOREM
a2 + b2 = c2
6 M
eter
s
b
62 + b2 = 102
36+ b2 = 100b2 = 64
b = √64 = 8 feet
EXAMPLE 4EXAMPLE 4
Eric & Emily have a 5 meter ladder. It must reach exactly 3 meters up a building, how far away from the building should the ladder be placed?
Eric & Emily have a 5 meter ladder. It must reach exactly 3 meters up a building, how far away from the building should the ladder be placed?
Step 1: Circle Important numbers.
Step 2: Draw a picture of the situation.STEP 3: PYTHAGOREAN THEOREM
a2 + b2 = c2
3 M
eter
s
b
32 + b2 = 52
9+ b2 = 25b2 = 16
b = √16 = 4 meters
Example 5 Tim rode 3 miles due north, then 3 miles due east. How far, to the nearest mile, is Tim from where he started?
Example 5 Tim rode 3 miles due north, then 3 miles due east. How far, to the nearest mile, is Tim from where he started?
Draw a diagram: Draw a diagram:
3
3x
Ex 5 Ex 5
32 + 32 = x2
9 + 9 = x2
18 = x2
x = √18 = 4.2
Tim is 4.2 miles from where he started.
Example 6 Mary rode 4 miles due north, then 4 miles due east. How far, to the nearest mile, is Mary from where he started?
Example 6 Mary rode 4 miles due north, then 4 miles due east. How far, to the nearest mile, is Mary from where he started?
Draw a diagram: Draw a diagram:
4
4x
Ex 6 Ex 6
42 + 42 = x2
16 + 16 = x2
32 = x2
x = √32 = 5.6
Mary is 5.6 miles from where he started.
Example 7 Town X is 30 miles directly west town Y. Town Z is 40 miles south of town Y. Tom lives in X and wants to get to Z. How much shorter is the direct route from town X to town Z than the longer route traveling through town Y?
Example 7 Town X is 30 miles directly west town Y. Town Z is 40 miles south of town Y. Tom lives in X and wants to get to Z. How much shorter is the direct route from town X to town Z than the longer route traveling through town Y?
X Y
Z
40 miles
30 miles
???
a2 + b2 = c2
302+402=c2
900 +1600 = c2
√2500 = c²50 = c
Did I answer question?
Ex 7Ex 7
Long route = 70 miles
Short route = 50 miles
70 – 50 = 20
20 miles shorter.
Long route = 70 miles
Short route = 50 miles
70 – 50 = 20
20 miles shorter.
A boat sails due East from a Harbour (H), to a marker buoy (B),3 miles away. At B the boat turns due South and sails for 2. miles to a Lighthouse (L). It then returns straight to harbour. What is the total distance travelled by the boat?
∴Total distance travelled = 3 + 2 + 3.6 = 8.6 miles
HB
L
3 miles
2 miles
Example 8
32+ 22 = c2
9 + 4 = c2
13 = c2
C2 = √13 = 3.6
ex 9 The length of a rectangle is 8 in, the length of its diagonal is 10 in. Find the area of the rectangle.ex 9 The length of a rectangle is 8 in, the length of its diagonal is 10 in. Find the area of the rectangle.
8 in
10 in b
Area of rectangleA = L x WWe need the width!
a2 + b2 = c2
82 + b2 = 102
64 + b2 = 100b2 = 100 – 64b2 = 36b = √36 = 6
area = 8 x 6 area = 48 sq inches
� Use your calculator for the rest� Use your calculator for the rest
EXAMPLE 10EXAMPLE 10� You're locked out of your house and the only open
window is on the second floor, 24 feet above the ground. You need to borrow a ladder from one of your neighbors. There's a bush along the edge of the house, so you'll have to place the ladder 7 feet from the house. What length of ladder do you need to reach the window?
� You're locked out of your house and the only open window is on the second floor, 24 feet above the ground. You need to borrow a ladder from one of your neighbors. There's a bush along the edge of the house, so you'll have to place the ladder 7 feet from the house. What length of ladder do you need to reach the window?
Step 1: Circle Important numbers.
Step 2: Draw a picture of the situation.
24 fe
et
7 feet
EXAMPLE 10EXAMPLE 10� You're locked out of your house and the only open
window is on the second floor, 24 feet above the ground. You need to borrow a ladder from one of your neighbors. There's a bush along the edge of the house, so you'll have to place the ladder 7 feet from the house. What length of ladder do you need to reach the window?
� You're locked out of your house and the only open window is on the second floor, 24 feet above the ground. You need to borrow a ladder from one of your neighbors. There's a bush along the edge of the house, so you'll have to place the ladder 7 feet from the house. What length of ladder do you need to reach the window?
24 fe
et
7 feet
STEP 3: PYTHAGOREAN THEOREMa2 + b2 = c2
242 + 72 = c2
576 + 49 = c2
625 = c2
C = √625 = 25 feet
Ex 11 On a regulation baseball field, the bases are 90 ft apart. How far is it on a direct line from 1st base to 3rd base?
3rd base 1st base
home plate
a2 + b2 = c2
902 + 902 = c2
8100 + 8100 = c2
16200 = c2
C2 = √16200 =127.3 ft
EXAMPLE 12EXAMPLE 12
Firefighters have a 40 foot extension ladder. In order to reach 28 feet up a building, how far away from the building should the foot of the ladder be placed?
Firefighters have a 40 foot extension ladder. In order to reach 28 feet up a building, how far away from the building should the foot of the ladder be placed?Step 1: Circle Important numbers.
Step 2: Draw a picture of the situation.STEP 3: PYTHAGOREAN THEOREM
a2 + b2 = c2
28 fe
et
b
282 + b2 = 402
784 + b2 = 1600b2 = 816
b = √816 = 28.6 feet
