review of roots 4 this is a collection of warm-ups and practice from class. 4 click to advance the...
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REVIEW OF ROOTS
This is a collection of warm-ups and practice from class.
Click to advance the slide and follow along. You can use the scroll bar at the right to
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WARM-UP
What is the prime factorization of the following numbers (ex. 14=2 • 7):9, 16, 25, 10, 17, 24, 27
What are the square roots of the following numbers:9, 16, 25, 10, 17, 24, 27
Answers 9 = 3•3 16 = 2•2•2•2 25 = 5•5 10 = 2•5 17 = 17 24 = 2•2•2•3 27 = 3•3•3
9 = 3
16 = 4
25 = 5
10 = 10 or 3.2
17 = 17 or 4.1
24 = 2 6 or 4.9
27 = 3 3 or 5.2
Chapter 10 – Right Triangles
Why should you care? LOTS OF STANDARDS: Students use the Pythagorean
theorem to determine distance and find missing lengths of sides of right triangles.
Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side.
Students know and are able to use angle and side relationships in problems with special right triangles, such as 30°, 60°, and 90° triangles and 45°, 45°, and 90° triangles.
Chapter 10 – Right Triangles
Why should you care? TRIGONOMETRY: You’ll need this stuff
for next year!! Mr. Taylor’s Opinion. Pythagorean theorem
and the trigonometric functions are EXTREMELY useful for DOING practical problems involving graphing(drawing) and measurement.
ROOTS – From Chapter 3
WHY NOT USE A CALCULATOR?– What is the square root of 5 on a calculator?
– approximately 2.236
– What’s the square root of 5 squared?
– 5
– What’s the 2.236 squared?
– 4.999696 (close to 5 but not exactly)
THEREFORE, if we want exactly the square root of 5, use 5
Finding exact roots
To simplify a number which includes a radical, find the prime factorization of the radicand and move all the perfect squares out front.
Examples. 4, 6, 8, 12, 15, 18
Pythagorean Theorem
The Pythagorean Theorem states that for any triangles with legs a and b and hypotenuse c that a2 + b2 = c2
1. If a right triangle has legs 3 and 4, what is the length of the hypotenuse?
2. If a right triangle has a leg 2 and a hypotenuse 10, what is the length of the other leg?
3. If a triangle has sides 5, 6, and 8 is it a right triangle?
Pythagorean Theorem
The Pythagorean Theorem states that for any triangles with legs a and b and hypotenuse c that a2 + b2 = c2
1. If a right triangle has legs 3 and 4, what is the length of the hypotenuse?
32 + 42 = c2
9 + 16 = c2
25 = c2
25 = c2
5 = c
Pythagorean Theorem
The Pythagorean Theorem states that for any triangles with legs a and b and hypotenuse c that a2 + b2 = c2
2. If a right triangle has a leg 2 and a hypotenuse 10, what is the length of the other leg?
22 + b2 = (10)2
4 + b2 = 10b2 = 10 – 4b2 = 6
b2 = 6
b = 6
Pythagorean Theorem
The Pythagorean Theorem states that for any triangles with legs a and b and hypotenuse c that a2 + b2 = c2
2. If a triangle has sides 5, 6, and 8 is it a right triangle?
52 + 62 = 82
25 + 36 = 6451 not equal 64NO, Not a righttriangle
WARM-UP
What are the lengths of the missing sides?
4590
45
7
7
72
3090
60 126
6362 + b2 = 122
b2 = 144 – 36
b = 108b = 6 3
72 + 72 = c2
c2 = 49+49
c = 2(7)(7)c = 7 2
Gold Boxes from p. 424
In a 45-45-90 Triangle, the measure of the hypotenuse is 2 times the leg.
4590
45
x
x
x2x2 + x2 = c2
c2 = x2 + x2 c2 = 2x2
c = 2x2
c = x 2
45-45-90 is an isosceles triangle
What about this?
4590
45
72
72
72 2 = 7 * 2 = 14
45-45-90 is an isosceles triangle
4590
45
76
76
76 2
= 72*2*3 = 7 * 2 * 3 = 143
45-45-90 is an isosceles triangle
4590
45
3
3
32
45-45-90 is an isosceles triangle
4590
45
Better
7
2
7
2
27
2
2
2
7
2
7
2
27This answerWould neverBe on a M.C.Test
45-45-90 is an isosceles triangle
4590
45
Better
8
2
8
242
28
2
2
2
8
2
8
24
Preview 30/60/90
30-60-90 is half an equilateral triangle 30-60-90 (Assume short side is opposite
small angle)
3090
60 8
4
43
42 + X2 = 82
X2 = 64 – 16
X = 48
X = 4 3
30/60/90 is half of an equilateral (60/60/60) triangle. Therefore, the side opposite the 30 will always be half of the side opposite the 90 and the side opposite the 90 will always be twice the side opposite the 30.
30-60-90 is half an equilateral triangle
3090
60 14
7
73
72 + X2 = 142
X2 = 196 – 49
X = 147X = 7 3
30-60-90 is half an equilateral triangle
3090
60 12
6
63
62 + X2 = 122
X2 = 144 – 36
X = 108X = 6 3
30-60-90 is half an equilateral triangle
3090
60 2m
m
m3
m2 + X2 = (2m)2
X2 = 4m2 – m2
X2 = 3m2
X = m 3
WARM-UP
What are the lengths of the missing sides?
4590
45
9
9
92
3090
60 105
5352 + b2 = 102
b2 = 100 – 25
b = 75b = 5 3
92 + 92 = c2
c2 = 81+81
c = 2(9)(9)c = 9 2
45-45-90 is an isosceles triangle
4590
45
9
9
92
45-45-90 is an isosceles triangle
4590
45
5
5
52
45-45-90 is an isosceles triangle
4590
45
62
62
62 2 = 6 * 2 = 12
45-45-90 is an isosceles triangle
4590
45
76
76
76 2 = 7 * 2 * 3 = 143
45-45-90 is an isosceles triangle
4590
45
3
3
32
45-45-90 is an isosceles triangle
4590
45
Better
7
2
7
2
27
2
2
2
7
2
7
2
27This answerWould neverBe on a M.C.Test
45-45-90 is an isosceles triangle
4590
45
Better
10
2
10
252
210
2
2
2
10
2
10
25
30-60-90 is half an equilateral triangle 30-60-90 (Assume short side is opposite
small angle)
3090
60 8
4
43
42 + X2 = 82
X2 = 64 – 16
X = 48
X = 4 3
30-60-90 is half an equilateral triangle
3090
60 14
7
73
72 + X2 = 142
X2 = 196 – 49
X = 147X = 7 3
30-60-90 is half an equilateral triangle
3090
60 2m
m
m3
m2 + X2 = (2m)2
X2 = 4m2 – m2
X2 = 3m2
X = m 3
Rules: In a 45-45-90 Triangle, the measure of the
hypotenuse is the leg times 2 In a 30-60-90 Triangle:
hypotenuse = 2 x shorter leglonger leg = 3 x shorter leg
To go in reverse direction, reverse the operation.– For instance, to go from hypotenuse to leg in a 45-45-
90, divide by 2 Answers must have no perfect squares under the
radicals and no radicals in the denominator.
30-60-90 is half an equilateral triangle
3090
60 426
213
2133
30-60-90 is half an equilateral triangle
3090
60 2
1
3
30-60-90 is half an equilateral triangle
3090
60 4
2
23
30-60-90 is half an equilateral triangle
3090
60
5
3
35
3
5 3
310
30-60-90 is half an equilateral triangle
3090
60
6
323
36
3
6
34
3
6