Question

Mr Diaz's class is solving the following problem. Given the points A(-2, -1), B(6,5), and DC-4,2) what must be the y-coordinate of point E if its x-coordinate is 2 and segment AB is congruent to segment DE? Which student is correct?

a) Student A: The y-coordinate of point E is 3.
b) Student B: The y-coordinate of point E is 7.
c) Student C: The y-coordinate of point E is -1.
d) Student D: The y-coordinate of point E is 1.

Answer 1

To find the y-coordinate of point E, we first need to find the equation of the line passing through points A and B. We can use the formula for the equation of a line, which is y = mx + b, where m is the slope and b is the y-intercept. By finding the equations of the lines AB and DE, we can determine the value of the y-coordinate for point E. The correct answer is Student D: The y-coordinate of point E is 2. Hence the correct answer is option D

Step-by-step explanation:

To find the y-coordinate of point E, we first need to find the equation of the line passing through points A and B. We can use the formula for the equation of a line, which is y = mx + b, where m is the slope and b is the y-intercept.

First, let's find the slope of the line AB. The slope is given by the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of two points on the line.

Using the coordinates of points A(-2, -1) and B(6, 5), we have m = (5 - (-1)) / (6 - (-2)) = 6 / 8 = 3 / 4.

Now that we have the slope, we can find the equation of the line AB. Using the point-slope form, y - y1 = m(x - x1), where (x1, y1) is a point on the line, we substitute (x1, y1) = (-2, -1) and m = 3 / 4:

y + 1 = (3 / 4)(x + 2)

y + 1 = (3 / 4)x + 3 / 2

y = (3 / 4)x + 1 / 2

Now that we have the equation of the line AB, we can find the equation of the line DE, which is congruent to AB. Since AB and DE are congruent, they have the same slope, which is 3 / 4.

Therefore, the equation of the line DE is y = (3 / 4)x + b, where b is the y-intercept. Since point E has an x-coordinate of 2, we can substitute x = 2 into the equation to find the y-coordinate of point E.

y = (3 / 4)(2) + b

y = 3 / 2 + b

Now, we need to find the value of b. To do this, we can use the fact that AB and DE are congruent, which means they have the same y-intercept. Since we know that point B has coordinates (6, 5), we can substitute these coordinates into the equation of the line AB to find the value of b:

5 = (3 / 4)(6) + b

5 = 9 / 2 + b

b = 5 - 9 / 2

b = 5 - 4.5 = 0.5

Finally, we substitute the value of b into the equation of the line DE to find the y-coordinate of point E:

y = 3 / 2 + 0.5

y = 2

Therefore, the correct answer is Student D: The y-coordinate of point E is 2.