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Question 556822: if the coordinates of the end point of ab are a(-6,3) b(2,-3) and the midpoint is (2.5,1) how do i find the length of ab
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Something is wrong in this problem. If the (x, y) coordinate points at the ends of the line ab are at:
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a = (-6,3) and
b = (2,-3)
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then the midpoint of the line ab is not located at (2.5, 1)
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So either one or both of the points a and b is wrong, or the midpoint is wrong. Or possibly all three are wrong.
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Here is the way to do this problem. Let's assume that the coordinates for a and b are correct. It will help you to visualize the following actions if you make a quick sketch of the coordinate axes and then plot the two points. You don't need to be accurate for this sketch. The purpose of the sketch is just to help you visualize what is going on. Notice that point a is in the second quadrant (the upper left quarter) of the coordinate axes where x is to the left of the y-axis by 6 units and y is 3 units above the x-axis. And point b is in the fourth quadrant (the lower right quarter) where b is 2 units to the right of the y-axis and 3 units below the x-axis.
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Starting with these two locations and only moving in the horizontal and vertical directions, how can we get from point a to point b. Let's start at point a and move horizontally to the right until we are above point b. [For reference, let's call this location point c. And point c should be (2, 3) just to make sure you are doing this correctly.] At that point we switch directions and go vertically down until we arrive at point b.
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Look at your sketch. Can you see that figure acb is a right triangle? Its legs are ac and cb and its hypotenuse is ab.
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How long is leg ac? You should see that ac goes from -6 on the x axis to +2 on the x-axis. Therefore, ac is 8 units long. (Notice that this distance can be found by subtracting the x coordinate of point a from the x coordinate of point b. That is by subtracting 2 - (-6) and getting 2 + 6 = 8.)
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Then how long is leg cb? This time you should be able to see that this vertical leg begins at 3 units above the x-axis and goes down to b at 3 units below the x-axis, a total distance of 6 units. (This time notice that this distance can be found by subtracting the y coordinate of point c from the y coordinate of point b. That is by subtracting -3 minus 3 and getting -3 -3 = -6 which is a length of 6 units downward.)
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So we have right triangle acb with legs ac = 8 and cb = 6. The length ab is the hypotenuse of this triangle and it is the distance you are trying to find.
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Since this is a right triangle shown on your sketch, you can use the Pythagorean theorem to find the length of the hypotenuse. Remember that the Pythagorean theorem says that the square of the hypotenuse is equal to the sum of the squares of the two legs. So letting H represent the hypotenuse (line ab), M represent line ac, and N represent the leg cb, we can write the Pythagorean theorem as:
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H^2 = M^2 + N^2
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Now we can substitute 8 for the leg M and 6 for the leg N to get:
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H^2 = 8^2 + 6^2
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8^2 is equal to +64 and 6^2 is equal to +36. So we can say:
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H^2 = 64 + 36
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and the right side of this equation totals to 100. So we have:
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H^2 = 100
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and we can solve for H by taking the square root of both sides to get:
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H = 10
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So the distance from point a to point b (the length of line ab) is equal to this hypotenuse and is 10 units in length. That's the answer you were to find. Notice that you did not need to know the midpoint of line ab to find this distance.
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Just for added measure, how do you find the midpoint of line ab? Not too tough.
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We begin by finding the x-value of the midpoint of line ac on your sketch. Recall that line ac started at x = -6 and went horizontally to x = +2. Recall that we decided this line was 8 units long. So its midpoint will be half of 8 units which will make it 4 units away from each of its ends. We either go 4 units to the right of -6 or 4 units to the left of +2. Either way we find that the value of x at the midpoint is x = -2. So our midpoint of line ab has the x value of -2.
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We continue by finding the y-value of the midpoint of line cb on your sketch. Recall that line cb started at y = +3 and went vertically to y = -3. Recall that we decided this line was 6 units long. So its midpoint will be half of 6 units or will be 3 units away from each end. We either go 3 units down from +3 or 3 units up from -3. Either way we find that the value of y at the midpoint is 0. So our midpoint of line cb has the y value of 0 and this is the y value for the midpoint of line ab.
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Now we can say that the midpoint of line ab is at (-2, 0)
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Hope this helps you to understand this problem. Maybe sketching out what is involved will help you to understand what you are doing when you work with the distance formula. You don't need to memorize the distance formula if you make a sketch such as we did in this problem and then apply the Pythagorean theorem to solve it.
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Good luck. With a little practice this will become easier to understand what you need to do.
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