Question

NO LINKS OR ANSWERING QUESTIONS YOU DON'T KNOW!!!

Point L is represented by the ordered pair (4, -3) and is shown on the coordinate plane below.

a. Point M is 5 units away from point L. Plot two different locations that could represent where point M is on the coordinate plane. List the coordinates below.

b. Point K is located at (-7, -3). What is the length of the line segment between point L and point K? Show or explain your answer using absolute value.

Answer 1

Part (a)

Answer: Point M could be at (9, -3) and could be at (-1, -3)

There are infinitely many answer choices

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Step-by-step explanation:

We start at point L(4,-3). We could move 5 units to the right. Doing so means we add on 9 to the x coordinate to get 4+5 = 9. That moves us to (9,-3) which is one possible location for point M.

Now move back to point L(4,-3). Let's move 5 units to the left and we'll get to (-1,-3) which is another possible location for M. I subtracted 5 from the x coordinate of (4,-3).

There are infinitely many places M can be. It turns out that M can be anywhere on the circle of radius 5 and centered at (4,-3). The equation of this circle is (x-4)^2 + (y+3)^2 = 25.

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Part (b)

Answer: 11 units

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Step-by-step explanation:

Point L is at (4, -3)

Point K is at (-7, -3)

Both have -3 as the y coordinate. So we focus on the distance between the x coordinates.

To find that x coordinate distance, we subtract and use absolute value like so

|x1-x2| = |4 - (-7)| = |4 + 7| = |11| = 11

Order doesn't matter with subtraction that involves absolute value

|x2-x1| = |-7-4| = |-11| = 11

Either way, we see that the distance from K to L is 11 units. This is the same as the length of segment KL.

Answer 2

a. M1 = (0, 0), M2 = (4, 2)

b. LK = |-7-4| = 11

Step-by-step explanation:

a. The infinite number of points that are 5 units from point M comprise a circle of radius 5 centered at M. There are 12 points on that circle that have integer coordinates.

Conveniently, 5 is the hypotenuse of a right triangle with sides 3 and 4. So, any point that has coordinate differences from L that are (±3, ±4) or (±4, ±3) will be 5 units away. (The signs can be chosen independently, so that's a total of 8 points.) Of course, points 5 units away on the same horizontal or vertical line will also be possible locations of point M: (4±5, -3) or (4, -3±5), a total of 4 more points.

The points we have shown in the attachment are (4, 2) and (0, 0).

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b. The distance formula can tell you the length of LK:

LK = √((-7-4)² +(-3-(-3))²) = √(-11)² = |-11| = 11

A length is always positive. The distance along a horizontal line can be found by subtracting one end-point x-coordinate from the other. In order to ensure the length is positive, the absolute value of the difference must be used.

In the case of the distance formula, we end up with √(d²) = |d|, where d is the difference of x-coordinates on the horizontal line.