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CD is perpendicular to AB and passes through point C(5, 12).

If the coordinates of A and B are (-10, -3) and (7, 14), respectively, the x-intercept of CD is {blank A}. The point {blank B} lies on CD.

Options:
Blank A:
(12,0)
(15,0)
(17,0)
(19,0)

Blank B:
(-5,24)
(-2,19)
(7,-10)
(8,11)

User Alsein
by
7.9k points

1 Answer

3 votes

Answer:

Blank A: (17, 0)

Blank B: (-2, 19)

Explanation:

Blank A:

Step 1: Find the slope of AB:

Before we can find the equation of CD, we'll first need to find the slope of AB

We can do this using the slope formula, which is given by:

m = (y2 - y1) / (x2 - x1), where

  • m is the slope,
  • (x1, y1) is one point,
  • and (x2, y2) is another point.

Thus, we can plug in (-10, -3) for (x1, y1) and (7, 14) for (x2, y2) in the slope formula to find m, the slope of AB:

m = (14 - (-3)) / (7 - (-10))

m = (14 + 3) / (7 + 10)

m = 17 / 17

m = 1

Thus, the slope of AB is 1.

Step 2: Find the slope of CD:

The slope of perpendicular lines are negative reciprocals of each other as shown by the following formula:

m2 = -1 / m1, where

  • m2 is the slope of the line we're trying to find,
  • and m1 is the slope of the line we know.

Thus, we can plug in 1 for m1 in the perpendicular slope formula to find m2, the slope of CD:

m2 = -1 / 1

m2 = -1

Thus, the slope of CD is -1.

Step 3: Find the y-intercept of CD:

One of the equations we can use when looking for intercepts is the slope-intercept form of a line, whose general equation is given by:

y = mx + b, where

  • (x, y) is any point on the line,
  • m is the slope,
  • and b is the y-intercept.

Thus, we can plug in (5, 12) for (x, y) and -1 for m to find b, the y-intercept of the line, allowing us to have the full equation in slope-intercept of CD:

12 = -1(5) + b

12 = -5 + b

17 = b

Thus, the equation of CD is y = -x + 17

For any x-intercept, the y-coordinate will always be 0 since the line is intersecting the x-axis.

Thus, we can find the x-coordinate of the x-intercept by plugging in 0 for y in y = -x + 17 and solving for x:

0 = -x + 17

-17 = -x

17 = x

Thus, the x-coordinate of the x-intercept of CD is 17.

Thus, the coordinates of the x-intercept of CD are (17, 0)

Blank B:

We can see that (-2, 19) lies on CD when we plug in (-2, 19) for (x, y) in y = -x + 17, as we get 19 on both sides of the equation when simplifying:

19 = -(-2) + 17

19 = 2 + 17

19 = 19

Thus, (-2, 19) lies on CD.

User Jittakal
by
8.1k points

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