AMC 10 · 2008 · #20
Grade 8 geometry-2dTrapezoid ABCD has bases AB and CD and diagonals intersecting at K. Suppose that AB=9, DC=12, and the area of △AKD is 24. What is the area of trapezoid ABCD?
Pick an answer.
AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A trapezoid ABCD has parallel bases AB = 9 and DC = 12. Its two diagonals cross at K, splitting the trapezoid into four triangles. One of those triangles, AKD, has area 24. Find the total area of the trapezoid.
Givens: ABCD is a trapezoid with AB parallel to DC.; AB = 9 and DC = 12.; The diagonals AC and BD meet at K.; The area of triangle AKD is 24.
Unknowns: The area of the whole trapezoid ABCD.
Understand
Restated: A trapezoid ABCD has parallel bases AB = 9 and DC = 12. Its two diagonals cross at K, splitting the trapezoid into four triangles. One of those triangles, AKD, has area 24. Find the total area of the trapezoid.
Givens: ABCD is a trapezoid with AB parallel to DC.; AB = 9 and DC = 12.; The diagonals AC and BD meet at K.; The area of triangle AKD is 24.
Plan
Primary tool: #7 Identify Subproblems
Secondary: #1 Draw a Diagram
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.
Execute — Answer: D
8.G.A.5 Step 1 Get the diagonal split ratio
- Because AB is parallel to DC, the top triangle KAB and the bottom triangle KCD have equal matching angles (alternate interior angles across the parallel lines, plus vertical angles at K), so they are similar.
- Their sizes scale like their parallel sides, 9 to 12, which is 3 to 4.
- That same 3:4 ratio is how K cuts each diagonal: AK:KC = 3:4 and BK:KD = 3:4.
💡 Parallel bases force the two end triangles into the same shape, so the short base to long base ratio is the ratio of every matching piece.
7.RP.A.2 Step 2 Scale up to triangle DKC
- Look at triangles AKD and DKC.
- They share vertex D, and their bases AK and KC sit on the same straight line AC, so both triangles have the same height dropped from D to that line.
- When triangles share a height, their areas are in the same ratio as their bases.
- So area(AKD):area(DKC) = AK:KC = 3:4.
- Since area(AKD) = 24, area(DKC) is 24 times 4/3, which is 32.
💡 Same height means area just tracks the base, so a longer base grows the area by the same factor.
5.NF.B.4 Step 3 Find the other two triangles
- Now use the split BK:KD = 3:4.
- Triangles AKB and AKD share vertex A with bases BK and KD on line BD, so their areas are in ratio 3:4, giving area(AKB) = 24 times 3/4 = 18.
- The same way, triangles BKC and DKC share vertex C with bases BK and KD, so area(BKC) = 32 times 3/4 = 24.
💡 The same shared-height rule turns each known triangle into its neighbor by one clean multiplication.
6.G.A.1 Step 4 Add the four pieces
- The four triangles tile the whole trapezoid, so add their areas: 18 for AKB, 24 for BKC, 32 for CKD, and 24 for DKA.
- The total is 98, so the area of trapezoid ABCD is 98, which is choice (D).
💡 Cutting a figure into triangles you can measure, then summing, rebuilds the whole area.
8.G.A.5 Because AB is parallel to DC, the top triangle KAB and the bottom triangle KCD h 7.RP.A.2 Look at triangles AKD and DKC. They share vertex D, and their bases AK and KC si 5.NF.B.4 Now use the split BK:KD = 3:4. Triangles AKB and AKD share vertex A with bases B 6.G.A.1 The four triangles tile the whole trapezoid, so add their areas: 18 for AKB, 24 Review
Reasonableness: 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.
Alternative: Place the trapezoid on coordinates, since the angles are free to choose: let D=(0,0), C=(12,0), A=(0,h), B=(9,h). Compute where diagonals AC and BD cross, use area(AKD)=24 to solve for the height h, then apply the trapezoid area formula (1/2)(9+12)h. It returns 98 as well.
CCSS standards used (min grade 8)
8.G.A.5Use informal arguments to establish facts about angles and the angle-angle criterion for similarity of triangles (Showing that the parallel bases make triangles KAB and KCD similar, so K divides each diagonal in the 3:4 ratio of the bases.)7.RP.A.2Recognize and represent proportional relationships between quantities (Using the fact that triangles sharing a height have areas proportional to their bases to scale the given area of 24.)5.NF.B.4Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction (Multiplying 24 by 4/3 and 3/4 to get the areas 32, 18, and 24 of the other triangles.)6.G.A.1Find the area of polygons by composing into rectangles or decomposing into triangles (Adding the four triangle areas to rebuild the total area of the trapezoid.)
⭐ 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.
⭐ 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.
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