AMC 10 · 2008 · #20

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Parallel-Cut Area Ratio
similar-trianglesarea-trianglesratio-proportion identify-subproblems ↑ Prerequisites: similar-triangles
📏 Medium solution 💡 2 insights
Problem
A trapezoid ABCD has parallel bases AB = 9 and DC = 12. Its two diagonals cross at K, splitting the trapezoid into four triangles. Of those four, triangle AKD has area 24. Find the total area of trapezoid ABCD.

Pick an answer.

(A)
$\ 92$
(B)
$\ 94$
(C)
$\ 96$
(D)
$\ 98$
(E)
$\ 100$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The diagonals cut the trapezoid into four triangles, so instead of chasing the whole area at once, find each triangle's area separately and add them. The parallel bases make the top and bottom triangles similar, which pins down the ratio in which K divides each diagonal. Once that 3:4 ratio is known, every triangle's area follows from the one given area of 24 by simple scaling.

1STEP 1

Get the diagonal split ratio

AB is parallel to DC, so triangles KAB and KCD are similar and K cuts both diagonals in ratio 3:4.

△ KAB ∼ △ KCD → AK/KC=BK/KD=AB/CD=9/12=3/4
2STEP 2

Scale up to triangle DKC

Triangles AKD and DKC share the height from D to line AC, so their areas scale by AK:KC = 3:4, giving DKC = 32.

[AKD]/[DKC]=AK/KC=3/4 → [DKC]=24·4/3=32
3STEP 3

Find the other two triangles

With BK:KD = 3:4 the same way, 24 shrinks to AKB = 18 and 32 shrinks to BKC = 24.

[AKB]=24·3/4=18, [BKC]=32·3/4=24
4STEP 4

Add the four pieces

The four pieces tile the trapezoid, so 18 + 24 + 32 + 24 = 98 is the area of ABCD, choice (D).

[ABCD]=[AKB]+[BKC]+[CKD]+[DKA]=18+24+32+24=98
Answer
98
The two 'side' triangles AKD and BKC both came out to 24, which is exactly the symmetry you expect in a trapezoid (they must be equal). The four pieces 18, 24, 32, 24 also line up with the similarity: top:bottom = 18:32 = 9:16 = (3/4)², the square of the side ratio, as similar triangles must. A total of 98 sits between the smallest choice 92 and largest 100, so it is a sensible size.
💡Key takeaway

Diagonals slice a trapezoid into four triangles, and the parallel-base ratio lets you grow one known area into all the rest, then just add.

  • Get the diagonal split ratio
  • Scale up to triangle DKC
  • Find the other two triangles
  • Add the four pieces

More like this

Same archetype — closest grade level first.