Use a Negative Reciprocal to Test a Right Angle

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: A nonvertical line has slope 2/3. Find the slope of a perpendicular line. Candidate answers: (A) -2/3 (B) -3/2 (C) 2/3 (D) 3/2.
• Independent check inputs: A nonvertical line has slope 3/4. Find the slope of a perpendicular line. Candidate answers: (A) -3/4 (B) -4/3 (C) 3/4 (D) 4/3.

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. Two nonvertical perpendicular lines have slope product negative one. Taking the negative reciprocal of a nonzero slope supplies the slope required for a perpendicular direction.

Here is the method on the supplied teaching case. For two nonvertical perpendicular lines, slope product is −1. The negative reciprocal of 2/3 is −3/2; their product is −1.

The reciprocal and sign change are both necessary. Horizontal and vertical lines form a special perpendicular pair outside this finite nonzero-slope formula; do not divide by a zero slope.

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.

Check your reasoning

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Take it further

Changed-input task: 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. Show the mathematical reason, not only the selected answer. Expected result: -10/9. For two nonvertical perpendicular lines, slope product is −1. The negative reciprocal of 9/10 is −10/9; their product is −1.

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