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8. EE. 6 Similar Triangles & Slope y=mx y=mx+b Similar Triangles If triangles are similar, their sides have the same ratios…. 2 6 So these are… similar triangles 6 1 = 12 2 2 1 = 4 2 3 = 6 1 2 12 4 3 All 3 sides have the same ratio… 6 Similar Triangles If triangles are similar, their sides have the same ratios…. 1 4 So these are… 12 similar triangles 4 1 = 12 3 1 1 = 3 3 1 = 3 1 3 1 All 3 sides have the same ratio… 3 3 Similar Triangles If triangles are similar, their sides have the same ratios…. 10 6 12 15 So these are… similar triangles 6 2 = 9 8 3 9 10 2 = 15 3 8 = 12 2 3 All 3 sides have the same ratio… E L I Rise = 6 M Rise = 4 S Run = 3 Run = 2 S to I 4 2 = M to E 2 6 1 3 = 2 1 Think about the rise and run between 2 points like similar triangles. Then it will make sense that no matter which points you pick on a line, the slope will always be the same ratio. H T A M Rise = 1 Run = 2 M to A 1 2 = M to H 1 3 2 6 = Rise = 3 1 2 Run = 6 Remember that the rise and run between 2 points are like similar triangles. So it makes sense that no matter which points you pick on a line, the slope will always be the same ratio. G R E Rise =Rise 12 = 6 A Run = 2 T Run = 4 R to A 6 2 = G to T 3 12 1 4 = 3 1 Rise and run between 2 points is like similar triangles. So, no matter which points you pick on a line, the slope will always be the same. To be a Direct Variation a line must… • Go through the origin • Be linear x Now let’s look at the equation for this line… How do you get from the x-value to the y-value? y 2 Multiply by 2 = 4 1 Multiply by 2 = 2 0 Multiply by 2 = 0 -1 Multiply by 2 = -2 -2 Multiply by 2 = -4 y = 2x This line is a direct variation! Goes through the origin To be a Direct Variation a line must… • Go through the origin • Be linear Now let’s look at the equation for this line… x How do you get from the x-value to the y-value? y 4 Multiply by ½ = 2 2 Multiply by ½ = 1 0 Multiply by ½ = 0 -2 Multiply by ½ = -1 -4 Multiply by ½ = -2 y = ½x This line is a direct variation! Goes through the origin To be a Direct Variation a line must… • Go through the origin • Be linear x Now let’s look at the equation for this line… How do you get from the x-value to the y-value? y 2 Multiply by 3 = 6 1 Multiply by 3 = 3 0 Multiply by 3 = 0 -1 Multiply by 3 = -3 -2 Multiply by 3 = -6 y = 3x This line is a direct variation! Goes through the origin Let’s look at the direct variation equations…. y = 2x y = ½x y = mx y = 3x Direct variation can always be in this format This line is linear but not a direct variation! Linear but not a Direct Variation •Linear •Does not go through the origin x Now let’s look at the equation for this line… How do you get from the x-value to the y-value? y -2 Multiply by -2, subtract 1= 3 -1 Multiply by -2, subtract 1= 1 0 Multiply by -2, subtract 1= -1 1 Multiply by -2, subtract 1= -3 y = -2x -1 Does not go through the origin This line is linear but not a direct variation! Linear but not a Direct Variation •Linear •Does not go through the origin x y 1 Multiply by 3, add 2 = 5 0 Multiply by 3, add 2 = 2 Multiply by 3, add 2 = -2 Multiply by 3, add 2 = -5 -1 -2 Now let’s look at the equation for this line… How do you get from the x-value to the y-value? y = 3x + 2 Does not go through the origin This line is linear but not a direct variation! Linear but not a Direct Variation •Linear •Does not go through the origin How do you get from the x-value to the y-value? y 2 Multiply by ½, add 4 = 5 0 Multiply by ½, add 4 = x -2 -4 Now let’s look at the equation for this line… 4 Multiply by ½, add 4 = 3 Multiply by ½, add 4 = 2 y = ½x + 4 Does not go through the origin Let’s look at the direct variation equations…. y = -2x -1 y = 3x + 2 y = mx + b y = ½x + 4 Linear but not direct variation can always be Written in this format. • Explain… • How are similar triangles and slope related? – What is the format of an equation that is a direct variation? – What is the format of an equation that is linear, but not a direct variation? Then click here to check your answers! Check your answers… – How are similar triangles and slope related? • Think about the rise and run between 2 points like similar triangles. Then it will make sense that no matter which points you pick on a line, the slope will always be the same ratio. – What is the format of an equation that is a direct variation? • y = mx – What is the format of an equation that is linear, but not a direct variation? • y = mx + b