Geometry & Figures

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

Decide which side the halving line crosses

In the figure, point C lies on the graph of the direct proportion y=3xy=3x and has xx-coordinate 44, and A is the point (10,0)(10, 0). Side CB is parallel to the xx-axis and side BA is perpendicular to it, so OABC is a trapezoid. If the graph of the direct proportion y=axy=ax divides the area of trapezoid OABC into two equal parts, find the constant aa. (O is the origin.)

x y C B A O 4 10 y = 3x
RatiosGeometryExpressions
Your answer
How to solve
Strategy Identify Subproblems — Break it into subproblems: coordinates, total area, which side the line crosses (compare the diagonal OB's triangle with half the area), then one area equation.
1STEP 1

Locate C and B

C is on y=3x at x=4, and B is level with C directly above A.

y = 3 × 4 = 12 → C(4, 12), B(10, 12)
2STEP 2

Area of the trapezoid

Parallel sides: OA =10 and CB =10-4=6; height 12.

1/2 × (10 + 6) × 12 = 96, half is 48
3STEP 3

Which side does the line hit?

The diagonal OB cuts off triangle OAB. Compare it with half the area.

1/2 × 10 × 12 = 60 ＞ 48
4STEP 4

Area equation on side AB

The line meets AB (where x=10) at D(10, 10a). Triangle OAD has base 10 and height 10a.

1/2 × 10 × 10a = 48 → 50a = 48 → a = 24/25
Answer
24/25
D= (10, 9.6) lies on AB between 0 and 12, and triangle OAD has area 1/2 × 10 × 9.6 = 48, half of 96. Also 24/25 ＜ 12/10, the slope of OB, as expected.
Takeaway

Before solving, test the easy line through a corner -- it tells you which side the answer lives on.

  • Locate C and B
  • Area of the trapezoid
  • Which side does the line hit?
  • Area equation on side AB
Where next?
Another one like thissuggested
12 practice problems
Same look, different logicsuggested
Constant xy fixes the triangle's areaCount one quadrant, then use point symmetry
Step back and solidifysuggested
A line through the origin sweeps between endpoints

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▶ Practice — 12 problems