Competition · AMC preparation · step 4 of 4
AMC 10 · 2022B · #2
Grade 8 geometry-2d
Pick an answer.
AMC 10 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The figure is already drawn for us, but the key is to mark on it what the words say: AP=3, PD=2, right angle at P. Tool #1 (Diagram) keeps the labels straight — we see that BP is exactly the height of the rhombus measured from the base AD. From there Tool #7 (Subproblems) splits the work into two clean pieces: first find BP using the right triangle APB, then plug base × height into the parallelogram-area rule.
Read the side length
Since P lies on AD, the side AD = AP + PD = 5, and a rhombus has all sides equal, so AB = 5 too.
Knowing that a rhombus has four equal sides is Grade 5 "classify two-dimensional figures by properties" — the label travels around the shape.
A rhombus has four equal sides, so one labelled length travels all the way around it.
▸ Why?
Equal sides face equal angles, so the four corners repeat the same shape.
▸ Why?
Its opposite sides stay a constant gap apart and never meet, which is what makes it a parallelogram.
Use the right triangle
In right triangle APB the legs are AP = 3 and BP, with hypotenuse AB = 5, so Pythagoras gives BP = 4.
The classic 3-4-5 right triangle — Grade 8 "Pythagorean theorem to find an unknown side". You can almost spot it without computing.
8.G.B.7Identify SubproblemsApply the area formula
A rhombus is a parallelogram, so area = base × height = AD × BP = 5 × 4 = 20.
A rhombus is a parallelogram, so area = base × height — Grade 6 "find area by composing shapes". The answer is (D).
6.G.A.1Identify SubproblemsThis AMC 10 problem only needs Grade 8 "3-4-5 right triangle" — once BP comes out to 4, the rhombus area is just 5 × 4 = 20.
- Read the side length
- Use the right triangle
- Apply the area formula
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