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Grade 10 · lesson 2 of 4

Locate Where Parallelogram Diagonals Meet

Parallelogram diagonals bisect one another.

About 20 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: A parallelogram diagonal has endpoints (-2,3) and (4,7). At what point do its diagonals meet? Candidate answers: (A) (1,5) (B) (1,7) (C) (2,10) (D) (3,5).
  • Independent check inputs: A parallelogram diagonal has endpoints (-3,4) and (5,8). At what point do its diagonals meet? Candidate answers: (A) (1,6) (B) (1,8) (C) (2,12) (D) (4,6).

Learn the idea

Geometric properties can be expressed through coordinates as well as diagrams. Parallelogram angle and diagonal theorems, midpoint coordinates, perpendicular slopes and perpendicular base-height relationships provide exact arguments about figures. Parallelogram diagonals bisect one another. Averaging the x-coordinates and the y-coordinates of one diagonal’s endpoints therefore locates the common meeting point.

Here is the method on the supplied teaching case. Parallelogram diagonals bisect each other. Their meeting point is the midpoint of the supplied diagonal: ((−2+4)/2,(3+7)/2)=(1,5).

Average each coordinate independently and retain negative signs. The midpoint theorem applies because the figure is a parallelogram; an arbitrary quadrilateral’s diagonals do not necessarily bisect one another.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: A parallelogram diagonal has endpoints (-2,3) and (4,7). At what point do its diagonals meet? Candidate answers: (A) (1,5) (B) (1,7) (C) (2,10) (D) (3,5). Solution: Parallelogram diagonals bisect each other. Their meeting point is the midpoint of the supplied diagonal: ((−2+4)/2,(3+7)/2)=(1,5). The final answer is (1,5). Average each coordinate independently and retain negative signs. The midpoint theorem applies because the figure is a parallelogram; an arbitrary quadrilateral’s diagonals do not necessarily bisect one another.

Your turn

Check your reasoning

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Helpful tips
  • Parallelogram diagonals bisect one another.
  • Average each coordinate independently and retain negative signs.
A parallelogram diagonal has endpoints (-3,4) and (5,8). At what point do its diagonals meet?

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

(1,6)

Parallelogram diagonals bisect each other. Their meeting point is the midpoint of the supplied diagonal: ((−3+5)/2,(4+8)/2)=(1,6).

Take it further

Changed-input task: A parallelogram diagonal has endpoints (-9,10) and (11,14). At what point do its diagonals meet? Candidate answers: (A) (1,12) (B) (1,14) (C) (10,12) (D) (2,24). Show the mathematical reason, not only the selected answer. Expected result: (1,12). Parallelogram diagonals bisect each other. Their meeting point is the midpoint of the supplied diagonal: ((−9+11)/2,(10+14)/2)=(1,12).

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More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. A parallelogram diagonal has endpoints (-4,5) and (6,9). At what point do its diagonals meet?
Need a hint?
  • Parallelogram diagonals bisect one another.
  • Average each coordinate independently and retain negative signs.
2. A parallelogram diagonal has endpoints (-5,6) and (7,10). At what point do its diagonals meet?
Need a hint?
  • Parallelogram diagonals bisect one another.
  • Average each coordinate independently and retain negative signs.
3. A parallelogram diagonal has endpoints (-6,7) and (8,11). At what point do its diagonals meet?
Need a hint?
  • Parallelogram diagonals bisect one another.
  • Average each coordinate independently and retain negative signs.

Your choices are temporary. Feedback appears after you check all the answers.

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