16Hit1600
AlgebraHardFind missing coordinate using perpendicular slope

Algebra practice question

On a diagram used in an ice-core climate record presentation, drilling sites are plotted in the xy-plane, where each unit represents 1 kilometer. A supply route is represented by the line with equation 2x - 6y = 9.

A second route is represented by a line that passes through the points (-3, 8) and (5, k) and is perpendicular to the supply route. What is the value of k?

Answer: -16

Solving 2x - 6y = 9 for y gives y = (1/3)x - 3/2, so the supply route has slope 1/3. A perpendicular line must have slope -3, the negative reciprocal. The slope through (-3, 8) and (5, k) is (k - 8)/(5 - (-3)) = (k - 8)/8. Setting (k - 8)/8 = -3 gives k - 8 = -24, so k = -16. A student who uses slope 1/3 (parallel instead of perpendicular) gets k = 32/3; one who uses -1/3 (sign change only) gets k = 16/3; one who uses 3 (reciprocal only) gets k = 32.

Common wrong answers

If you answered 32
You took the reciprocal of 1/3 but forgot the negative sign: with slope 3, (k-8)/8 = 3 gives k = 32. Perpendicular slopes are negative reciprocals, so the slope is -3 and k = -16.
If you answered 32/3
You used the supply route's own slope, 1/3, which makes the second line parallel, not perpendicular. Flip and negate to get slope -3, which gives k = -16.
If you answered 16/3
You changed the sign of 1/3 but never flipped it. The negative reciprocal of 1/3 is -3, not -1/3, so (k-8)/8 = -3 and k = -16.
If you answered -24
That's k - 8, not k. After finding k - 8 = -24, add 8 to get k = -16.

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