AMC 10 · 2009 · #18

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
coordinate-geometrysimilar-trianglesarea-triangles convert-to-algebra ↑ Prerequisites: coordinate-geometry
📏 Medium solution 💡 3 insights
Problem
In rectangle ABCD the sides are AB=8 and BC=6. Point M is the middle of the diagonal AC. Point E sits on side AB so that segment ME meets the diagonal at a right angle. Find the area of triangle AME.

Pick an answer.

(A)
$\frac{65}{8}$
(B)
$\frac{25}{3}$
(C)
9
(D)
$\frac{75}{8}$
(E)
$\frac{85}{8}$

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The rectangle's corners are right angles, so tool #4 (Introduce a Variable) lets me drop the figure onto a coordinate grid, put corner A at the origin, and name the unknown point E=(e,0) on side AB. Tool #1 (Draw a Diagram) pins down where M and E sit so the right angle at M is easy to see. Tool #7 (Identify Subproblems) splits the task into three clean pieces: first the length AM from the diagonal, then the length AE from the right angle, then the area from those two lengths.

1STEP 1

Place the rectangle on a grid

Set A=(0,0), B=(8,0), C=(8,6), D=(0,6); then AC's midpoint is M=(4,3) and E on AB is (e,0).

A=(0,0), B=(8,0), C=(8,6), D=(0,6), M=(4,3), E=(e,0)
2STEP 2

Get AM from the diagonal

In right triangle ABC the legs 8 and 6 give AC=82+62=10AC=\sqrt{8^2+6^2}=10, and the midpoint halves it to AM=5.

AC=√(8²+6²)=10, AM=1/2 AC=5
3STEP 3

Get AE from the right angle

AME and ABC share angle A and each holds a right angle, so they are similar: AMAB=AEAC\frac{AM}{AB}=\frac{AE}{AC} gives AE=254AE=\frac{25}{4}.

AM/AB=AE/AC → AE=10·5/8=25/4
4STEP 4

Compute the area of triangle AME

Base AE lies flat on the x-axis, so the height is M's y-coordinate 3: the area is 12⋅254⋅3\frac{1}{2}\cdot\frac{25}{4}\cdot 3 = 758\frac{75}{8}, choice (D).

[△ AME]=1/2 · AE · 3=1/2·25/4·3=75/8 → (D)
Answer
75/8
The area 75/8=9.375 is just above choice (C) 9, which fits a triangle a bit larger than a 9-unit reference. As a sanity check on the shape: triangle AME is similar to ABC with ratio AM/AB=5/8, so its area should be (5/8)² times [△ ABC]=1/2·8·6=24. That gives 25/64·24=75/8, the same value, so the two methods agree.
💡Key takeaway

Half the diagonal is 5; the little right triangle is a shrunk copy of the big one, so scale 5/8 gives AE=25/4 and area 75/8.

  • Place the rectangle on a grid
  • Get AM from the diagonal
  • Get AE from the right angle
  • Compute the area of triangle AME

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