Download 5.7 The Pythagorean Theorem

5.7 The Pythagorean Theorem
Objectives:
G.SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems.
For the Board: You will be able to use the Pythagorean Theorem and its converse and its inequalities
to solve problems.
Bell Work 5.7:
Classify each triangle by its angle measures.
1.
40°
80°
70°
3. Simplify 2 3 ∙ 2 3 .
4. If a = 6, b = 7, and c = 12, find a2 + b2 then find c2. Which is greater, a2 + b2 or c2?
Anticipatory Set:
Vocabulary:
The sides which intersect at the right angle are called the legs of
the right triangle. Since they are usually different lengths,
they are referred to as the short leg and the long leg.
The side across or opposite the right angle is called the
hypotenuse.
short leg
(a)
Hypotenuse
(c)
long leg
(b)
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.
a2 + b2 = c2
Open the book to page 361 and read example 1 part A.
Example: Find the value of x. Give your answer in simplest radical form.
Do not approximate your answer with a calculator.
a.
b.
6
2
15
x
x
2 2 + 6 2 = x2
4 + 36 = 40 = x2
x = 40  4  10  2 10
5
x 2 + 52 = 152
x2 + 25 = 225
x2 = 225 – 25 = 200
x = 200  100  2  10 2
White Board Activity:
Practice: Find the value of x. Give your answer in simplest radical form.
Do not approximate your answer with a calculator.
8
a.
b.
4
24
x
x
4
42 + 82 = x2
16 + 64 = x2
x2 = 80
x  80  2  2  2  2  5
 4 4 5  22 5  4 5
x2 + 42 = 242
x2 + 16 = 576
x2 = 560
x  560  2  2  2  2  5  7
 4 4 35  2  2 35  4 35
Open the book to page 361 and read example 2.
Example: Randy is building a rectangular picture frame. He wants the ratio
of the length to the width to be 3:1 (use 3x for the length and x
for the width) and the diagonal to be 12 cm.
How wide should the frame be? Round to the nearest tenth of a
centimeter.
x2 + (3x)2 = 122
x2 + 9x2 = 144
10x2 = 144
x2 = 14.4
x = 3.8 cm
12
3x
x
White Board Activity:
Practice: How high will a 30 foot ladder reach if it is placed 5 feet from the base of the building?
Round to the nearest tenth of a foot.
52 + x2 = 302
25 + x2 = 900
x2 = 875
x ft 30 ft
x = 29.6 ft
5 ft
A Pythagorean Triple is a set of three positive integers a, b, and c that satisfy the equation
c2 = a2 + b2.
3, 4, 5 is an example of a Pythagorean Triple.
32 + 42 = 9 + 16 = 25 = 52
0.5, 1.2, 1.3 is not an example of a Pythagorean Triple, even through 0.52 + 1.22 = 1.32, because these
numbers are not positive integers.
2, 3, 4 is not an example of a Pythagorean Triple, because 22 + 32 does not equal 42.
White Board Activity:
Practice: Which of the following are Pythagorean Triples?
a. 5, 12, 13
b. 4, 5, 9
2
2
5 + 12 = 25 + 144 = 169 = 132
Yes
42 + 52 = 16 + 25 = 41 which is not 92( 81)
82 + 152 = 64 + 225 = 289 = 172
Yes
c. 8, 15, 17
No
Open the book to page 362 and read example 3.
Example: Find the missing side length, then tell if the side lengths form a Pythagorean triple.
a.
b.
4
14
12
48
142 + 482 = 196 + 2304 = c2
c2 = 2500
c = 50
Yes
42 + b2 = 122
b2 = 128
No
16 + b2 = 144
b = 128  2  64  8 2
White Board Activity:
Practice: Find the missing side length, then tell if the side lengths form a Pythagorean triple.
a.
b.
x2 + 242 = 262
24
10
8
x2 = 576 = 676
x2 = 100
26
2
2
2
8 + 10 = x
x = 10
2
64 + 100 = x
yes
2
164 = x
x = 12.8
no
c.
d.
1
12
30
162 + 302 = x2
256 + 900 = x2
1156 = x2
x = 34
Yes
16
2.4
+ 2.42 = x2 1 + 5.76 = x2
6.76 = x2
x = 2.6
No
Pythagorean Inequalities Theorem
In ΔABC, c is the length of the longest side.
If c2 < a2 + b2, then ΔABC is an acute triangle.
If c2 = a2 + b2, then ΔABC is a right triangle.
If c2 > a2 + b2, then ΔABC is an obtuse triangle.
a
c
b
a
c
b
c
a
b
Recall: Triangle Inequality Theorem
The sum of the measures of the two smaller sides must be larger than the measure of the
third side.
Example: 4, 8, 9 form a triangle because 4 + 8 = 12 > 9
3, 4, 7 do not form a triangle because 3 + 4 = 7 which is not > 7
2, 5, 12 do not form a triangle because 2 + 5 = 7 which in not > 12
Open the book to page 363 and read example 4.
Example: Tell if the measures con be the side lengths of a triangle. If so classify the triangle as acute,
obtuse, or right.
A. 7, 8, 9
B. 8, 8, 15
7 + 8 = 15 > 9
8 + 8 = 16 > 15
Yes, triangle.
Yes, triangle.
92
72 + 82
152
8 2 + 82
81
49 + 64
225
64 + 64
81
113
225
128
81 < 113, the triangle is acute.
225 > 128,the triangle is obtuse.
White Board Activity:
Practice: Tell if the measures can be the side lengths of a triangle. If so classify the triangle as acute,
obtuse, or right.
a. 5, 7, 10
b. 5, 8, 17
5 + 7 = 12 > 10
5 + 8 = 13 < 17
Yes they form a triangle
No does not form a triangle
2
2
2
10
5 +7
100
25 + 49 = 74
Since 100 > 74 the triangle is obtuse
Since practice prob. b does not form a triangle, change one of the sides so that it does form a triangle.
Then determine whether it is acute, obtuse, or right.
Example: Change the 8 to 15, then 5 + 15 = 20 which is > 17.
172
52 + 152
289
25 + 225 = 250
Since 289 > 250 the triangle is obtuse.
Assessment:
Student pairs will complete “CHECK IT OUT” prob. 1 – 4 from this section.
Independent Practice:
Text: pgs. 364-367 prob. 2, 3, 6, 7, 9-11, 15, 16, 20, 21-24.
Extra Credit: pg. 365 prob. 30, 32, 34.