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Common Core Standards: Math

Grade 8

Geometry 8.G.B.8

8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Finding the length of a line segment can be incredibly easy. If we plot the two points on graph paper and find that the points form either a horizontal or a vertical line, all we have to do is count the number of boxes between the two points. Easier than finding a needle in a haystack with a giant magnet.

When the points form a slanted line, it gets a little tricky. Counting boxes isn't really an option, measuring the distance isn't exact enough, and there's no exact formula to find the distance between the two points…is there?

Your students should already be nice and comfortable with the Pythagorean Theorem and using the equation a2 + b2 = c2. If they aren't, make sure to explain that whole deal before moving on to how we can use it to calculate the distance between two points.

"What?" they'll ask. "What do two points have to do with right triangles? What witchcraft is this?"

It's not witchcraft. Actually, it's not even that hard. Students should understand that any two points that form a diagonal line can be thought of as the hypotenuse of a right triangle. If they don't believe you, draw in the horizontal and vertical legs of the triangle and then taunt them with an I-told-you-so dance. (We're just kidding about the I-told-you-so dance. That wouldn't be nice.)

Now, when given points in a coordinate system, students can imagine right triangles and calculate the distance using the Pythagorean Theorem. They'll still need to count the horizontal and vertical boxes to get the lengths of the legs—until they learn the distance formula, which is derived from the Pythagorean Theorem.

While you don't have to teach them the distance formula just yet, introducing them to the distance formula could be really useful, not to mention a fun and challenging glimpse into the future. (Now, that would be witchcraft.)

Drills

  1. What is the distance between (2, 4) and (2, -4)?

    Correct Answer:

    8

    Answer Explanation:

    If we actually draw the two points, we'll see that they form a vertical line. (See how the x coordinates are the same?) We can count the squares or realize that subtracting the y values would give us the distance: 4 – (-4) = 4 + 4 = 8. So our two points are 8 units apart. And we didn't even need the Pythagorean Theorem for that!


  2. What's the distance between (0, -3) and (9, -3)?

    Correct Answer:

    9

    Answer Explanation:

    Since the points share the same y coordinate, they form a horizontal line. There's no vertical dimension to it, so the x coordinates are all we need to worry about. Simply subtract the x coordinates to find the horizontal distance: 9 – 0 = 9. So the two points are 9 units apart. End of story.


  3. What is the distance between (1, 4) and (4, 8)?

    Correct Answer:

    5

    Answer Explanation:

    From one point to the other, we move 4 – 1 = 3 units horizontally and 8 – 4 = 4 units vertically. In other words, the legs of our right triangle are 3 and 4. Using a2 + b2 = c2, we end up with 32 + 42 = 9 + 16 = 25 = c2, or c = 5. Since we use the hypotenuse to represent the distance between the two endpoints, we know that our two points are 5 units apart. That's really all it takes.


  4. What is the distance between (-1, 1) and (5, 9)?

    Correct Answer:

    3

    Answer Explanation:

    When we draw our lines, we get legs of 5 – (-1) = 6 and 9 – 1 = 8. After substituting these numbers into the Pythagorean Theorem (62 + 82 = 36 + 64 = 100 = c2), we get a hypotenuse of c = 10. That's the distance between the two points as well. See how useful the Pythagorean Theorem can be?


  5. What is the distance between (-3, -2) and (2, 10)?

    Correct Answer:

    17

    Answer Explanation:

    Finding the horizontal distance gives us 2 – (-3) = 5 units and the vertical distance is 10 – (-2) = 12. With two legs of such different sizes, this triangle probably walks a little funny. Using the Pythagorean Theorem (52 + 122 = 25 + 144 = 169 = c2) gives us a hypotenuse of c = 13. Once again, we're saved by a right triangle and its best bud, Pythagoras.


  6. What is the distance between (-4, -6) and (4, 9)?

    Correct Answer:

    17

    Answer Explanation:

    First, we count how many vertical and horizontal squares we move to get from one point to another. Or, rather than count, we can subtract the coordinates: 4 – (-4) = 8 and 9 – (-6) = 15. In other words, we move 8 squares horizontally and 15 squares vertically, so the legs of our right triangle are 8 and 15. We don't know about your friends, but our good friend Pythagoras says that makes a hypotenuse of 17. (Pythagoras has his own theorem. How many of your friends can say that?)


  7. The distance between (0, 0) and another point is 20. What are the coordinates of the second point?

    Correct Answer:

    (12, 16)

    Answer Explanation:

    To find the horizontal and vertical distances of the legs of our right triangle, we usually subtract our x and y coordinates. This time, one of our coordinates is (0, 0), so we don't have to! Our only remaining coordinate is basically going to tell us the lengths of the legs. In other words, do the squares of both the x and y coordinates sum up to 202 or 400? This is only true for (B), since 122 + 162 = 144 + 256 = 400.


  8. The distance between (-3, 4) and another point is 15. What are the coordinates of the second point?

    Correct Answer:

    (5, 14)

    Answer Explanation:

    It's pretty hard to pick out the right answer just by looking, and working backwards from a2 + b2 = 152 leaves us stumped as to what to do next. On the other hand, we can compare the distances of the x and y coordinates from the given point and our choices. Taking the points in (A), 82 + 102 = 164 ≠ 225 = 152, so that can't be right. Answer choice (B), however, works because 92 + 122 = 225 = 152. If we do the same for (C) and (D)—just to be sure—we get 164 ≠ 225 and 157 ≠ 225.


  9. Which two points are separated by a distance of 17 units?

    Correct Answer:

    (-3, 11) and (5, -4)

    Answer Explanation:

    Yeah, you have to find the distance between each of those pairs of points. There's really no way around it. At least by now, you're well versed enough in the Pythagorean Theorem and all its a2 + b2 = c2 goodness to find hypotenuses (and distances as a result) quickly and easily! The distances we have to choose from are 20, , 17, and , respectively. Obviously, the only answer that's 17 is (C), so it has to be the right answer.


  10. What is the distance between any two points (x1, y1) and (x2, y2)? Use the Pythagorean Theorem and express your answer in terms of these coordinate values.

    Correct Answer:

    Answer Explanation:

    "Use the Pythagorean Theorem" is practically a giveaway. Since our distance is c in a2 + b2 = c2, solving for c gives us \sqrt{a^2+b^2}. Now, we can replace those a's and b's with coordinates…but how? We know the horizontal distance is the difference between the x coordinates and the vertical distance is the difference between the y coordinates. That being the case, we can replace a and b with x2 – x1 and y2 – y1, which gives us (B) as our answer. Of the other options, (A) is the midpoint formula, (C) is the slope formula, and (D) is an incorrect version of the distance formula.