16Hit1600
AlgebraHardFind x-intercept of a perpendicular line

Algebra practice question

On a desert botany expedition, a survey plot is drawn in the xy-plane, where each unit represents 10 meters. A sampling transect is represented by a line whose x-intercept is (9, 0) and whose y-intercept is (0, -6). A second transect is represented by the line that is perpendicular to the first transect and passes through the point (0, -6).

What is the x-coordinate of the x-intercept of the line representing the second transect?

Answer: -4

The first transect has slope (-6 - 0)/(0 - 9) = 2/3, so the perpendicular transect has slope -3/2. Since it passes through (0, -6), its equation is y = -(3/2)x - 6. Setting y = 0 gives (3/2)x = -6, so x = -4. A student who reuses the slope 2/3 (parallel instead of perpendicular) gets x = 9; one who uses -2/3 (sign change only) gets x = -9; one who uses 3/2 (reciprocal only) gets x = 4.

Common wrong answers

If you answered 4
You flipped 2/3 to 3/2 but kept it positive. Perpendicular slopes are negative reciprocals, so use -3/2 and the x-intercept is -4.
If you answered 9
You reused the first transect's slope of 2/3, which gives a parallel line — that just reproduces the original x-intercept. The perpendicular slope is -3/2, giving x = -4.
If you answered -9
You negated 2/3 but didn't flip it. The negative reciprocal of 2/3 is -3/2, not -2/3, so the x-intercept is -4.
If you answered -6
That's the y-intercept the line passes through, not the x-intercept. Set y = 0 in y = -(3/2)x - 6 to get x = -4.

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