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Question
Triangle ABC and triangle DEG are similar right triangles. Which proportion can be used to show that the slope of AC is equal to the slope of DG?
Responses
A 4−(−7)0−4
= −1−104−8
4 − ( − 7 ) 0 − 4 = − 1 − 10 4 − 8
B −4−(−7)0−4
= 4−8−1−(−10)
− 4 − ( − 7 ) 0 − 4 = 4 − 8 − 1 − ( − 10 )
C 0−4−4−(−7)
= −1−(−10)−4−8
0 − 4 − 4 − ( − 7 ) = − 1 − ( − 10 ) − 4 − 8
D 0−4−4−(−7)
= −4−8−1−(−10)
D- 0−4−4−(−7) = −4−8−1−(−10)
To show that the slope of AC is equal to the slope of DG in similar right triangles ABC and DEG, you can use the proportion:
A. 4 - (-7) / 0 - 4 = -1 - 10 / 4 - 8
To determine which proportion can be used to show that the slope of AC is equal to the slope of DG, we need to analyze the given information about the similar right triangles ABC and DEG.
In similar triangles, corresponding sides are proportional. The slope of a line can be calculated by dividing the change in the y-coordinate by the change in the x-coordinate.
Let's denote the coordinates of the points A, B, C, D, E, and G as (x₁, y₁), (x₂, y₂), (x₃, y₃), (x₄, y₄), (x₅, y₅), and (x₆, y₆) respectively.
Given that triangles ABC and DEG are similar right triangles, we can write the following ratios for the corresponding sides:
BC/EG = AC/DG
To find the slope of a line, we can use the formula:
slope = (change in y-coordinate)/(change in x-coordinate)
Therefore, we need to find the coordinates (x, y) for points A, C, D, and G.
Using the given information, we have:
A = (x₁, y₁),
C = (x₃, y₃),
D = (x₄, y₄),
G = (x₆, y₆)
To find the slope of line AC, we calculate:
slope_AC = (y₃ - y₁)/(x₃ - x₁)
To find the slope of line DG, we calculate:
slope_DG = (y₆ - y₄)/(x₆ - x₄)
By comparing the slopes, we can determine if they are equal.
Now, let's analyze the given response options A, B, C, and D:
A: 4−(−7)/0−4 = −1−10/4−8
B: −4−(−7)/0−4 = 4−8−1−(−10)
C: 0−4−4−(−7) = −1−(−10)−4−8
D: 0−4−4−(−7) = −4−8−1−(−10)
We compare the ratios in the given options with the ratios we derived earlier, BC/EG = AC/DG. The correct proportion should support this relationship.
Analyzing the options, it appears that option A:
4−(−7)/0−4 = −1−10/4−8
is the correct proportion that shows the equality of the slopes of AC and DG.
Therefore, the correct response is A.