AMC 10 · 2025 · #6

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
coordinate-geometryquadratic-equationsslope-intercept convert-to-algebra ↑ Prerequisites: quadratic-equations
📏 Medium solution 💡 2 insights
Problem
The line y = (1/3)x + 1 divides the square region 0 ≤ x ≤ 2 and 0 ≤ y ≤ 2 into an upper region and a lower region. The vertical line x = a divides the lower region into two regions of equal area. Then a can be written as √(s) - t, where s and t are positive integers. Find the value of s + t.

Pick an answer.

(A)
18
(B)
19
(C)
20
(D)
21
(E)
22

AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The unknown a is exactly the number to name and chase. The plan is: first draw the picture to see that the lower region is a trapezoid, then find its total area, then write the area of the left piece (from x=0 to x=a) as an expression in a, set that equal to half the total, and solve the resulting equation. Naming a turns 'cut the area in half' into an equation we can solve, and the answer comes out in the √(s) - t form the problem asks for.

1STEP 1

Draw the region and name its corners

The line goes from (0,1) to (2, 5/3), so the region below it is a trapezoid with corners (0,0), (2,0), (2, 5/3), (0,1).

corners: (0,0), (2,0), (2,5/3), (0,1)
2STEP 2

Find the total lower area

Averaging the parallel sides 1 and 5/3 over width 2 gives total area 8/3, so each piece must be 4/3.

A_total = 1/2(1 + 5/3)(2) = 8/3, 1/2A_total = 4/3
3STEP 3

Write the left piece's area in terms of a

The left piece is a trapezoid with heights 1 and (1/3)a + 1 over width a, so 1/2(2 + a/3)a = 4/3.

1/2(2 + a/3)a = 4/3
4STEP 4

Turn it into a clean equation

Expanding gives a + a²/6 = 4/3; multiplying by 6 clears the fractions and leaves a² + 6a - 8 = 0.

a + a²/6 = 4/3 → a² + 6a - 8 = 0
5STEP 5

Solve for a and read off the answer

Completing the square, (a+3)² = 17, so a = √(17) - 3; matching √(s) - t gives s + t = 20, choice (C).

(a+3)² = 17 → a = √(17) - 3, s + t = 17 + 3 = 20
Answer
20
Check that a = √(17) - 3 ≈ 1.12 sits between 0 and 2: it does, so the vertical cut really falls inside the region. It also lands just past the middle, which makes sense because the region is slightly taller on the right, so the left half has to be a bit wider to hold the same area. Plugging back, the left area is 1/2(2 + 1.12/3)(1.12) ≈ 1.33 = 4/3, exactly half of 8/3. With s = 17 and t = 3, s + t = 20 is choice (C).
💡Key takeaway

Turn 'cut the area in half' into an equation by naming the cut a, write the left trapezoid's area, set it to half the total, and solve to get a = √(17) - 3.

  • Draw the region and name its corners
  • Find the total lower area
  • Write the left piece's area in terms of a
  • Turn it into a clean equation
  • Solve for a and read off the answer

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