Download The Pythagorean Theorem

Unit 2: Right Triangles
SMART Packet #4
Identifying sides and angles of a right triangle
Student:
Teacher:
Standards
A.A.45
Determine the measure of a third side of a right triangle using the
Pythagorean theorem, given the lengths of any two sides
G.G.48
Investigate, justify, and apply the Pythagorean Theorem and its converse
The Pythagorean Theorem
The Pythagorean Theorem:
In a right triangle, if a and b are the measures of
the legs, and c is the hypotenuse, then a2 + b2 = c2
c
a
b
Example 1:
Find the length of the hypotenuse if a = 9 and b = 12.
a2 + b2 = c2
92 + 122 = c2
81 + 144 = c2
225 = c2
Substitute the given values into the Pythagorean Theorem
Simplify.
Take the square root of both sides
225  c 2
15 = c
To find 225 , press:
2nd
x2
2
2
5
ENTER
PRACTICE, PART I
Find the length of the missing side.
1. a = 7, b = 24, c = ?
a2 + b2 = c 2
(
)2 + (
)2 = (
)2
2. a = 8, b = 15, c = ?
3. a = 12, b = 5, c = ?
4.
5.
Example 2:
Find the length of the leg, to the nearest tenth, if a = 4 and c = 10.
a2 + b2 = c2
42 + b2 = 102
16 + b2 = 100
–16
–16
b2 = 84
Substitute the given values into the Pythagorean Theorem
Solve for b.
Take the square root of both sides.
b 2  84
b = 9.165...
Rounding to the nearest tenth:
b = 9.2
9.165...
.
T
E
N
T
H
S
6>5
The number to the right of
the tenths place is greater
than 5, so round 1 up to 2.
PRACTICE, PART II
Find the length of the missing side. Round decimals to the nearest tenth.
6. a = 3, b = ?, c = 5
7. a = ?, b = 7, c = 10
8.
9.
Example 3:
A 12-ft ladder is placed 5 ft from a building. How tall is the building, to the nearest tenth?
12 ft
5 ft
Substitute the values into the Pythagorean Theorem & solve for the missing side.
a2 + b2 = c2
52 + b2 = 122
25 + b2 = 144
b2 = 119
b = 10.9
The building is 10.9 feet tall.
PRACTICE, PART III: Regents Prep (with pictures)
10. The NuFone Communications Company must run a telephone line between two
poles at opposite ends of a lake, as shown in the diagram. The length and
width of the lake are 75 feet and 30 feet, respectively.
What is the distance between the two poles, to the nearest foot?
(1) 105
(2) 69
(3) 81
(4) 45
11. Tanya runs diagonally across a rectangular field that has a length of 40 yards
and a width of 30 yards. What is the length of the diagonal, in yards, that Tanya
runs?
(1) 50
(2) 70
(3) 60
(4) 80
12. Don placed a ladder against the side of his house as shown below.
(a) Find the value of x, the distance from the foot of the ladder to the base of
the house
(b) Simplify the equations below. Which one gives you the same value you
found for x in part (a)? That is your answer!
(1) x = 20 – 19.5
2
2
(2) x  20  19.5
(3) x = 202 – 19.52
2
2
(4) x  20  19.5
13. The diagram shows a kite that has been secured to a stake in the ground with a
20-foot string. The kite is located 12 feet from the ground, directly over point X.
What is the distance, in feet, between the stake and point X?
14. A wall is supported by a brace 10 feet long, as shown. If one end of the brace
is placed 6 feet from the base of the wall, how many feet up the wall does the
brace reach?
PRACTICE, PART IV: Regents Prep. (without pictures)
 Draw a picture AND label it with the given information.
15.A woman has a ladder that is 13 feet long. If she sets the base of the ladder on
level ground 5 feet from the side of a house, how many feet above the ground
will the top of the ladder be when it rests against the house?
16. The length of the hypotenuse of a right triangle is 34 inches and the length of
one of its legs is 16 inches. What is the length, in inches, of the other leg of this
right triangle?
(1) 16
(3) 25
(2) 18
(4) 30
17. An 18-foot ladder leans against the wall of a building. The base of the ladder is
9 feet from the building on level ground. How many feet up the wall, to the
nearest tenth of a foot, is the top of the ladder?
18. If the length of a rectangular television screen is 20 inches and its height is 15
inches, what is the length of its diagonal, in inches?
(1) 5
(2) 25
(3) 13.2
(4) 35