Coordinate Geometry and Conics · Lesson 1

Distance, midpoint and slope

Distance. The distance between (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is (x2−x1)2+(y2−y1)2\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. It is the Pythagorean theorem in disguise: the horizontal change and the vertical change are the legs of a right triangle, and the distance is its hypotenuse.

Midpoint. Average the coordinates: (x1+x22,y1+y22)\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right). Slope is rise over run, m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}. Subtract in the same order on top and bottom: if the second point's y comes first in the numerator, the second point's x comes first in the denominator.

Missing endpoint. If M is the midpoint of RS‾\overline{RS} and you know R, the step from R to M repeats from M to S. In one line: S=(2xM−xR,2yM−yR)S = (2x_M - x_R, 2y_M - y_R). Averaging R and M does not help; that only finds the middle of RM‾\overline{RM}.

Worked example Easy

In the standard (x, y) coordinate plane, what is the distance between the points (−2,3)(-2, 3) and (4,11)(4, 11)?

  1. 2172\sqrt{17}
  2. 1010Answer
  3. 1414
  4. 100100

How to solve it

  1. Horizontal change: 4−(−2)=64 - (-2) = 6.
  2. Vertical change: 11−3=811 - 3 = 8.
  3. Distance: 62+82=36+64=100=10\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10.

Why each choice is right or wrong

  • A. Incorrect. 217=682\sqrt{17} = \sqrt{68} comes from using 4−2=24 - 2 = 2 as the horizontal change, which drops the negative sign of −2-2. Subtracting −2-2 adds 2, so the change is 6.
  • B. Correct. The changes are 4−(−2)=64 - (-2) = 6 and 11−3=811 - 3 = 8, so the distance is 36+64=10\sqrt{36 + 64} = 10. Faster: 6, 8, 10 is the 3-4-5 right triangle doubled.
  • C. Incorrect. 14 adds the two changes, 6+86 + 8. The changes are the legs of a right triangle, so square them, add, and then take the square root.
  • D. Incorrect. 100 is 62+826^2 + 8^2 before the square root is taken. The distance is 100=10\sqrt{100} = 10.

Worked example Medium

In the standard (x, y) coordinate plane, point M at (1,−2)(1, -2) is the midpoint of RS‾\overline{RS}. Point R is at (−3,4)(-3, 4). What are the coordinates of point S?

  1. (−5,8)(-5, 8)
  2. (−1,1)(-1, 1)
  3. (4,−6)(4, -6)
  4. (5,−8)(5, -8)Answer

How to solve it

  1. From R to M, x changes by 1−(−3)=41 - (-3) = 4 and y changes by −2−4=−6-2 - 4 = -6.
  2. S is the same step past M: x=1+4=5x = 1 + 4 = 5 and y=−2+(−6)=−8y = -2 + (-6) = -8.
  3. Check: the midpoint of (−3,4)(-3, 4) and (5,−8)(5, -8) is (22,−42)=(1,−2)\left(\frac{2}{2}, \frac{-4}{2}\right) = (1, -2).

Why each choice is right or wrong

  • A. Incorrect. (−5,8)(-5, 8) subtracts in the wrong order, R−2MR - 2M: −3−2=−5-3 - 2 = -5 and 4−(−4)=84 - (-4) = 8. The endpoint is 2M−R=(5,−8)2M - R = (5, -8).
  • B. Incorrect. (−1,1)(-1, 1) averages R and M, which gives the midpoint of RM‾\overline{RM}, not the other endpoint. M is already the middle, so S lies beyond it.
  • C. Incorrect. (4,−6)(4, -6) is the step from R to M, the change in each coordinate. Add that step to M to reach S: (1+4,−2−6)=(5,−8)(1 + 4, -2 - 6) = (5, -8).
  • D. Correct. The step from R to M is 4 right and 6 down; the same step from M lands on (5,−8)(5, -8). Shortcut: S=(2⋅1−(−3),2(−2)−4)=(5,−8)S = (2 \cdot 1 - (-3), 2(-2) - 4) = (5, -8).

Trap Mixed order and the averaging reflex

In a slope, keep one point first on top and bottom. 5−(−1)3−(−1)\frac{5 - (-1)}{3 - (-1)} mixes the points and gets the sign wrong. For a missing endpoint, don't average the two points you know: the midpoint is given, so step past it.

Calculator When a calculator isn’t faster

Distance, midpoint and slope are one-line formulas, so hand work is usually quicker. If you want a check, type the arithmetic into your calculator, such as sqrt((4+2)^2+(11-3)^2), or plot the two points in the built-in Desmos calculator if you test online.