Competition · AMC preparation · step 4 of 4
AMC 8 · 2005 · #9
Grade 8 geometry-2d
Pick an answer.
AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #1 (Draw a Diagram) shows that the diagonal AC slices the quadrilateral into two triangles, △ ABC and △ ADC. Only one of them carries useful given information: △ ADC has two known sides (DA = CD = 17) and the angle between them (60°). The sides AB and BC at vertex B are a distraction. Tool #7 (Identify Subproblems) reduces the original quadrilateral question to a single subproblem about △ ADC: find its third side AC. The isosceles structure plus the 60° apex angle forces all three angles to be 60°, making the triangle equilateral, so AC = 17 with no calculation beyond the angle sum.
Draw the diagonal
Draw the diagonal AC; it splits ABCD into △ ABC and △ ADC, and every given measurement sits inside △ ADC.
Picking the triangle that carries the given side lengths and angle is the Grade 4 "classify a figure by its known properties" move.
4.G.A.2Draw A DiagramFocus on the isosceles triangle
In △ ADC two sides are equal (DA = CD = 17), so it is isosceles and its two base angles ∠ DAC = ∠ DCA.
"Two equal sides force two equal opposite angles" is the Grade 4 isosceles fact in its simplest form.
4.G.A.2Identify SubproblemsUse the angle sum
By the 180° angle sum, x + x + 60° = 180°, so each base angle x = 60°.
The 180° triangle-angle-sum fact is the Grade 8 "informal arguments about triangle angles" standard.
8.G.A.5Identify SubproblemsConclude the triangle is equilateral
All three angles are 60°, so △ ADC is equilateral — every side equal, giving AC = DA = CD = 17.
Recognizing the equiangular-equals-equilateral fact is the Grade 4 "classify 2-D figures" payoff.
Triangle ADC is equilateral, so its third side — the diagonal AC — is exactly as long as each of the two given equal sides DA and CD.
▸ Why?
All three angles of triangle ADC are 60 degrees, and a triangle whose angles are all equal has all sides equal.
▸ Why?
The angle at D is given as 60 degrees, and the other two angles each come out to 60 degrees as well.
▸ Why?
Because the two sides DA and CD are equal, the angles facing them — the angle at A and the angle at C — must be equal to each other.
▸ Why?
The three angles add to 180 degrees, so with the angle at D taking 60, the two equal remaining angles split the other 120 evenly, making each one 60.
▸ Why?
When all three angles are equal, each side faces an equal angle, so the side opposite the angle at D matches the sides opposite the angles at A and C.
The diagonal AC lives inside two triangles, but only one of them carries the given 17, 17, and 60°. That isosceles triangle with a 60° apex angle has to be equilateral, so the third side just equals 17.
- Draw the diagonal
- Focus on the isosceles triangle
- Use the angle sum
- Conclude the triangle is equilateral
A parent dashboard for the family lives at sensimlab.com.
More like this
Same problem type. Tags show what each one shares with this problem.
- AMC 8 2003 #6 Gr 8Similar level
Same sub-type: Weighted Group AggregationSame techniqueIdentify SubproblemsGiven the areas of the three squares in the figure, what is the area of the interior triangle?
- AMC 8 2007 #14 Gr 8Similar levelSame sub-type: Weighted Group AggregationSame techniqueIdentify Subproblems
The base of isosceles △ ABC is 24 and its area is 60. What is the length of one of the congruent sides?
- AMC 8 2010 #16 Gr 8Similar levelSame sub-type: Weighted Group AggregationSame techniqueIdentify Subproblems
A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle…