LEMMA

Lines & systems

Coordinate geometry: midpoint, distance, endpoints

ALEKS placement

Coordinates tell you where. Distance tells you how far apart.

What this covers

d=√(x2​−x1​)2+(y2​−y1​)2​,M=(2x1​+x2​​,2y1​+y2​​)
  • Read (x, y) as a location: x across, then y up or down.
  • For distance, find the horizontal gap and vertical gap by subtracting matching coordinates. These are the two short sides of a right triangle.
  • The straight distance is the long side: square each gap, add, then take the positive square root.
  • For midpoint, average the x-values and average the y-values.
  • To find a missing endpoint, set the midpoint formula equal to the known midpoint.

Worked example

Worked example

Find the distance between (-2, 3) and (4, 11).

  1. 4−(−2)=6First look across: from negative two to four is six units right. Subtract the starting location; do not add the two locations.
  2. 11−3=8Now look up: from three to eleven is eight units up. Six across and eight up form the short sides of a right triangle.
  3. d2=62+82The direct distance d is the slanted side. For a right triangle, its square equals the sum of the two short sides squared.
  4. d2=36+64=100Square means multiply a number by itself: six times six is thirty-six, and eight times eight is sixty-four. Add those squares.
  5. d=10What nonnegative number times itself is 100? It is 10. The points are 10 units apart; the answer is one length, not a new point.
Another worked example

Find the midpoint and the distance between A=(−1, −2) and B=(3, 1).

  1. M=(2−1+3​,2−2+1​)=(1,−21​)Average the x-coordinates, then average the y-coordinates separately.
  2. d2=(3−(−1))2+(1−(−2))2=42+32=25The changes, 4 across and 3 up, are the short sides of a right triangle.
  3. d=55 × 5 = 25, and a distance is never negative.

A common mistake

A mistake Lemma catches

From d2=36+64

d=6+8d=10

“A square root does not split over a sum: √36 + √64 = 6 + 8 = 14 answers a different question. Add first, then take the root: √(36 + 64) = √100 = 10.”

Try one

Sample problem

Find the midpoint of (-2, 2) and (4, 6).

(−2,2),(4,6)

Practice coordinates free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

Before this

After this