The Life Starter
Sign up
Grade 10 · unit review

Review: Geometry: Quadrilaterals and Coordinate Reasons

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.

Download reviews and answer keys

Bring the unit skills together

Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. A parallelogram has one interior angle 55°. Find an adjacent interior angle.
Need a hint?
  • Opposite parallelogram sides are parallel.
  • Choose adjacent rather than opposite angles for the 180-degree relation.
2. A parallelogram diagonal has endpoints (-7,8) and (9,12). 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 nonvertical line has slope 7/8. Find the slope of a perpendicular line.
Need a hint?
  • Two nonvertical perpendicular lines have slope product negative one.
  • The reciprocal and sign change are both necessary.
4. A triangle has vertices (0,0), (14,0), and (0,8). Find its area.
Need a hint?
  • A triangle with legs along the coordinate axes has an explicit right angle at their meeting point.
  • A slanted side is not automatically the height.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. A parallelogram has one interior angle 61°. Find an adjacent interior angle. Candidate answers: (A) 118 ° (B) 119 ° (C) 120 ° (D) 238 °. 2. 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). 3. A nonvertical line has slope 9/10. Find the slope of a perpendicular line. Candidate answers: (A) -10/9 (B) -9/10 (C) 10/9 (D) 9/10. 4. A triangle has vertices (0,0), (18,0), and (0,10). Find its area. Candidate answers: (A) 180 square units (B) 89 square units (C) 90 square units (D) 91 square units.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. 119 °. Adjacent angles in a parallelogram are supplementary because opposite sides are parallel. Thus the adjacent angle is 180−61=119°. Opposite angles, rather than adjacent ones, are equal. 2. (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). 3. -10/9. For two nonvertical perpendicular lines, slope product is −1. The negative reciprocal of 9/10 is −10/9; their product is −1. 4. 90 square units. The axes meet at a right angle, so the base 18 and perpendicular height 10 give A=(1/2)bh=90. The triangle is half its bounding rectangle.

A different response can also meet the criteria. Compare the evidence and reasoning, rather than copying the wording.