7-1 Ratios and Proportions I CAN Write a ratio Write a ratio expressing the slope of a line. Solve a...
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Transcript of 7-1 Ratios and Proportions I CAN Write a ratio Write a ratio expressing the slope of a line. Solve a...
![Page 1: 7-1 Ratios and Proportions I CAN Write a ratio Write a ratio expressing the slope of a line. Solve a linear proportion Solve a quadratic proportion Use.](https://reader035.fdocuments.in/reader035/viewer/2022062311/5a4d1b107f8b9ab05998f149/html5/thumbnails/1.jpg)
7-1 Ratios and ProportionsI CAN•Write a ratio•Write a ratio expressing the slope of a line.•Solve a linear proportion•Solve a quadratic proportion•Use a proportion to determine if a figure has been dilated.
![Page 2: 7-1 Ratios and Proportions I CAN Write a ratio Write a ratio expressing the slope of a line. Solve a linear proportion Solve a quadratic proportion Use.](https://reader035.fdocuments.in/reader035/viewer/2022062311/5a4d1b107f8b9ab05998f149/html5/thumbnails/2.jpg)
A ratio compares two numbers by division. The ratio
of two numbers a and b can be written as a to b, a:b,
or , where b ≠ 0. For example, the ratios 1 to 2,
1:2, and all represent the same comparison.
Example: There are 11 boys and 15 girls in class. Write the ratio of girls to boys.
15 to 11 15:11 1511
The order of the numbers matters!
![Page 3: 7-1 Ratios and Proportions I CAN Write a ratio Write a ratio expressing the slope of a line. Solve a linear proportion Solve a quadratic proportion Use.](https://reader035.fdocuments.in/reader035/viewer/2022062311/5a4d1b107f8b9ab05998f149/html5/thumbnails/3.jpg)
Writing Ratios to Express Slope of a Line
In Algebra I, you learned that the slope of a line (m) is an example of a ratio. Slope is a rate of change and can be expressed in the following ways:
rise run
yx
y2 – y1
x2 – x1
![Page 4: 7-1 Ratios and Proportions I CAN Write a ratio Write a ratio expressing the slope of a line. Solve a linear proportion Solve a quadratic proportion Use.](https://reader035.fdocuments.in/reader035/viewer/2022062311/5a4d1b107f8b9ab05998f149/html5/thumbnails/4.jpg)
Writing Ratios to Express Slope of a Line
Write a ratio expressing the slope of the give line.
Substitute the given values.
Simplify.
![Page 5: 7-1 Ratios and Proportions I CAN Write a ratio Write a ratio expressing the slope of a line. Solve a linear proportion Solve a quadratic proportion Use.](https://reader035.fdocuments.in/reader035/viewer/2022062311/5a4d1b107f8b9ab05998f149/html5/thumbnails/5.jpg)
Ratios in Similar PolygonsA ratio can involve more than two numbers. For the rectangle, the ratio of the side lengths may be written as 3:7:3:7.
![Page 6: 7-1 Ratios and Proportions I CAN Write a ratio Write a ratio expressing the slope of a line. Solve a linear proportion Solve a quadratic proportion Use.](https://reader035.fdocuments.in/reader035/viewer/2022062311/5a4d1b107f8b9ab05998f149/html5/thumbnails/6.jpg)
Example : Using Ratios
The ratio of the side lengths of a triangle is 4:7:5, and its perimeter is 96 cm. What is the length of the shortest side?
Let the side lengths be 4x, 7x, and 5x.
4x + 7x + 5x = 96 16x = 96 x = 6
The length of the shortest side is 4x = 4(6) = 24 cm.
![Page 7: 7-1 Ratios and Proportions I CAN Write a ratio Write a ratio expressing the slope of a line. Solve a linear proportion Solve a quadratic proportion Use.](https://reader035.fdocuments.in/reader035/viewer/2022062311/5a4d1b107f8b9ab05998f149/html5/thumbnails/7.jpg)
A proportion is an equation stating that two ratios are equal to each other.
In a proportion, the cross products ad and bc are equal.
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Solving Linear Proportions
To solve a proportion, “CROSS MULTIPLY AND SIMPLIFY.”
Example
4 = k10 65
10k = 260 Cross multiply
10k = 260 10 10
Simplify by dividing both sides of equation by 10
k = 26
![Page 9: 7-1 Ratios and Proportions I CAN Write a ratio Write a ratio expressing the slope of a line. Solve a linear proportion Solve a quadratic proportion Use.](https://reader035.fdocuments.in/reader035/viewer/2022062311/5a4d1b107f8b9ab05998f149/html5/thumbnails/9.jpg)
Example
3 = 4(x + 3) (x + 8)
3(x + 8) = 4(x + 3) Cross multiply
3x + 24 = 4x + 12 Simplify by distributing-3x -3x Get variable on same side of equation
24 = x + 12-12 – 12 12 = x
Solving Linear Proportions
![Page 10: 7-1 Ratios and Proportions I CAN Write a ratio Write a ratio expressing the slope of a line. Solve a linear proportion Solve a quadratic proportion Use.](https://reader035.fdocuments.in/reader035/viewer/2022062311/5a4d1b107f8b9ab05998f149/html5/thumbnails/10.jpg)
Your Turn
7 = 2 3x (x + 4)
x = -28
Solving Linear Proportions
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Solving Quadratic Proportions
Example
2y = 8 9 4y
8y2 = 72 Cross multiply
8 8
y2 = 9
392
yy
Simplify
Take the positive and negative square root of both sides
![Page 12: 7-1 Ratios and Proportions I CAN Write a ratio Write a ratio expressing the slope of a line. Solve a linear proportion Solve a quadratic proportion Use.](https://reader035.fdocuments.in/reader035/viewer/2022062311/5a4d1b107f8b9ab05998f149/html5/thumbnails/12.jpg)
Solving Quadratic ProportionsYour Turn
14 = 2x x 7
7x
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Solving Quadratic ProportionsExample
(x+3) = 9 4 (x+3)
(x+3)(x+3) = 36 Cross multiply
x2 + 6x + 9 = 36 FOIL -36 -36
x2 + 6x – 27 = 0 Solve quadratic equations by setting equation = 0
( x – 3 )( x + 9 ) = 0 Factorx -3 = 0 x + 9 =0
x = 3 x = -9
Use Zero Product Property to find solutions
![Page 14: 7-1 Ratios and Proportions I CAN Write a ratio Write a ratio expressing the slope of a line. Solve a linear proportion Solve a quadratic proportion Use.](https://reader035.fdocuments.in/reader035/viewer/2022062311/5a4d1b107f8b9ab05998f149/html5/thumbnails/14.jpg)
Solving Quadratic ProportionsYour Turn
(x – 4) = 20 5 (x – 4)
x = 14 x = -6
![Page 15: 7-1 Ratios and Proportions I CAN Write a ratio Write a ratio expressing the slope of a line. Solve a linear proportion Solve a quadratic proportion Use.](https://reader035.fdocuments.in/reader035/viewer/2022062311/5a4d1b107f8b9ab05998f149/html5/thumbnails/15.jpg)
Solving Quadratic ProportionsExample
3 = (x – 8)(x + 9) (3x – 8)
3(3x – 8) = (x – 8)(x + 9)
9x – 24 = x2 + 9x – 8x – 72
9x – 24 = x2 + x – 72
– 9x + 24 – 9x + 24
0 = x2 – 8x – 48 0 = (x – 12)(x + 4)
x – 12 = 0 x + 4 = 0x = 12 or x = – 4
![Page 16: 7-1 Ratios and Proportions I CAN Write a ratio Write a ratio expressing the slope of a line. Solve a linear proportion Solve a quadratic proportion Use.](https://reader035.fdocuments.in/reader035/viewer/2022062311/5a4d1b107f8b9ab05998f149/html5/thumbnails/16.jpg)
Dilations and Proportions
When a figure is dilated, the pre-image and image are proportional.
You can use proportions to find missing measures and to check dilations!
Refer to the “Dilations as Proportions” Worksheet in your Unit plan.We will now work examples 1 and 2.
![Page 17: 7-1 Ratios and Proportions I CAN Write a ratio Write a ratio expressing the slope of a line. Solve a linear proportion Solve a quadratic proportion Use.](https://reader035.fdocuments.in/reader035/viewer/2022062311/5a4d1b107f8b9ab05998f149/html5/thumbnails/17.jpg)
Dilations as Proportions
C U
TE
8 cm3 cm
U G
LY
7.5 cm
Ex) Rectangle CUTE was dilated to create rectangle UGLY. Find the length of LY.
3 = 87.5 UG
Pre-image and image of dilated figures are proportional
3 = 87.5 LY
Opposite sides of a rectangle are congruent.
3LY = 8(7.5)
3LY = 60
LY = 20 cm
Cross multiply
Simplify
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Dilations as Proportions
6 in.
2.25 in.
20 in.
10 in.
8 in.
3 in.
30 in.
5 in.
A B C D
Ex) Determine which of the following figures could be a dilation of the triangle onthe right (There could be more than one answer!)
16 in.
6 in.
Triangle A
6 = 2.2516 6
36 = 2.25(16)?36 = 36? YES
Triangle B
20 = 10 16 6
20(6) = 10(16)?120 = 160? NO
Triangle C
8 = 3 16 6
8(6)=16(3)?48 = 48?YES
Triangle D
30 = 5 16 6
30(6) =16(5)?180 = 80? NO
![Page 19: 7-1 Ratios and Proportions I CAN Write a ratio Write a ratio expressing the slope of a line. Solve a linear proportion Solve a quadratic proportion Use.](https://reader035.fdocuments.in/reader035/viewer/2022062311/5a4d1b107f8b9ab05998f149/html5/thumbnails/19.jpg)
Now complete #1 & 2 on Dilations as Proportions Worksheet
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