Geometry & Figures

Depth · one concept from first principles · step 2 of 4

Find which side the bisector hits first

A trapezoid-shaped quadrilateral sits on the x-axis with the origin on its bottom side. Find the line y=ax through the origin that splits its area into two equal parts.
x y O A(-6, 6) B(-6, 0) C(2, 0) D(2, 2)
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Your answer
How to solve
Strategy Identify Subproblems — Break it into subproblems. First the total area (it is a trapezoid, since AB and DC are both vertical). Then decide which side the line exits through, by testing the line through vertex A: if the piece it cuts off is already too big, the true line must hit side AB below A. Finally a single triangle area gives the exit point, and the slope follows.
1STEP 1

Total area

AB and DC are vertical, so ABCD is a trapezoid with parallel sides 6 and 2 and width 2-(-6)=8.

1/2×(6+2) × 8=32
2STEP 2

Test the line through A

The line through O and A cuts off triangle ABO with base BO=6 and height 6.

1/2 × 6 × 6=18 ＞ 16
3STEP 3

Find the exit point on AB

Let the line meet AB at P(-6, k). The left piece is triangle PBO with base BO=6 and height k.

1/2 × 6 × k=16 → k=16/3
4STEP 4

Slope through the origin

The line y=ax passes through P(-6, 16/3).

16/3=-6a → a=-8/9
Answer
y=-(8/9)x
Left piece: triangle with base 6 and height 16/3, area 16. Right piece: 32-16=16. Equal, as needed. The slope -8/9 is a bit flatter than the slope -1 of line OA, matching the test in step 2.
Takeaway

Before cutting a shape in half, find out which side the cut lands on -- testing a corner tells you right away.

  • Total area
  • Test the line through A
  • Find the exit point on AB
  • Slope through the origin
Where next?
Level upsuggested
Bisect a pentagon by a line from one vertexPivot a line past a triangle's extreme vertices
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12 practice problems
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A line through the origin sweeps between endpoints

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▶ Practice — 12 problems