AMC 10 · 2007 · #14
Grade 8 geometry-2dA triangle with side lengths in the ratio 3 : 4 : 5 is inscribed in a circle with radius 3. What is the area of the triangle?
Pick an answer.
AMC 10 2007 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 triangle whose side lengths are in the ratio $3:4:5$ is drawn inside a circle of radius $3$ so that all three corners touch the circle. Find the area of this triangle.
Givens: The triangle's sides are in the ratio $3:4:5$; The triangle is inscribed in a circle (its three vertices lie on the circle); The circle has radius $3$; Answer choices: (A) 8.64, (B) 12, (C) $5\pi$, (D) 17.28, (E) 18
Unknowns: The actual area of the inscribed triangle
Understand
Restated: A triangle whose side lengths are in the ratio $3:4:5$ is drawn inside a circle of radius $3$ so that all three corners touch the circle. Find the area of this triangle.
Givens: The triangle's sides are in the ratio $3:4:5$; The triangle is inscribed in a circle (its three vertices lie on the circle); The circle has radius $3$; Answer choices: (A) 8.64, (B) 12, (C) $5\pi$, (D) 17.28, (E) 18
Plan
Primary tool: #1 Draw a Diagram
Secondary: #5 Look for a Pattern, #4 Introduce a Variable, #7 Identify Subproblems
The whole problem turns on one picture, so Tool #1 (Draw a Diagram) is primary: sketching the triangle inside the circle exposes the key fact that its longest side is a diameter. Getting there needs Tool #5 (Look for a Pattern) to recognize $3:4:5$ as the famous Pythagorean triple, which makes the triangle a right triangle. Then Tool #4 (Introduce a Variable) sets a single scale factor $k$ so the ratio sides $3,4,5$ become real lengths, and Tool #7 (Identify Subproblems) splits the job into clean stages — prove it is right, find the diameter, scale the legs, then take the area.
Execute — Answer: A
8.G.B.6 Step 1 Recognize the 3-4-5 right triangle
- The sides are in the ratio $3:4:5$, and $3^2+4^2=9+16=25=5^2$.
- By the converse of the Pythagorean theorem, any triangle whose sides satisfy this is a right triangle, with the longest side ($5$ in the ratio) opposite the right angle.
- So this inscribed triangle is right-angled.
💡 Seeing $9+16=25$ instantly flags the classic right-triangle triple hiding in the ratio.
7.G.B.4 Step 2 The hypotenuse is a diameter
- Draw the right triangle with all three corners on the circle.
- A right angle that sits on a circle always opens onto a diameter — the side across from the right angle stretches straight through the center.
- That side is the hypotenuse, the one matching the $5$ in the ratio.
- Since the radius is $3$, the diameter is $2\times 3 = 6$, so the hypotenuse has length $6$.
💡 The right angle points across to the widest span of the circle, which is a diameter.
7.G.A.1 Step 3 Scale the ratio up to real lengths
- Let $k$ be the factor that turns the ratio numbers into true side lengths, so the sides are $3k$, $4k$, and $5k$.
- The $5k$ side is the hypotenuse, which we just found equals $6$.
- Solving $5k = 6$ gives $k = \tfrac{6}{5} = 1.2$.
- The two legs are then $3k = 3.6$ and $4k = 4.8$.
💡 One unknown stretch factor rescales the whole shape once you pin down a single real length.
6.G.A.1 Step 4 Take half of leg times leg
- In a right triangle the two legs are perpendicular, so they act as base and height.
- The area is half their product.
- Using the legs $3.6$ and $4.8$: area $=\tfrac12(3.6)(4.8) = \tfrac12(17.28) = 8.64$.
- That matches choice (A).
💡 Perpendicular legs already give you a base and a matching height, so no extra work is needed.
8.G.B.6 The sides are in the ratio $3:4:5$, and $3^2+4^2=9+16=25=5^2$. By the converse o 7.G.B.4 Draw the right triangle with all three corners on the circle. A right angle that 7.G.A.1 Let $k$ be the factor that turns the ratio numbers into true side lengths, so th 6.G.A.1 In a right triangle the two legs are perpendicular, so they act as base and heig Review
Reasonableness: The circle's area is $\pi r^2 = 9\pi \approx 28.3$, and the triangle sits inside it, so its area must be well under $28$, and $8.64$ is comfortably less. A sharper check: the plain $3\text{-}4\text{-}5$ triangle has area $\tfrac12(3)(4)=6$, and scaling every length by $k=1.2$ multiplies area by $k^2=1.44$, giving $6\times 1.44 = 8.64$ — the same answer two ways. The trap answer (D) $17.28$ is exactly $3.6\times 4.8$, i.e. forgetting the factor of $\tfrac12$; (C) $5\pi\approx 15.7$ wrongly expects a $\pi$ (areas of triangles carry no $\pi$); (B) $12$ and (E) $18$ don't come from any correct step.
Alternative: Skip the individual legs and scale areas directly. The base $3\text{-}4\text{-}5$ triangle has area $6$. The real triangle is similar to it with side ratio $\tfrac{6}{5}$, and areas of similar figures scale as the square of the side ratio: $\left(\tfrac{6}{5}\right)^2 = \tfrac{36}{25} = 1.44$. So the area is $6 \times 1.44 = 8.64$, matching (A).
CCSS standards used (min grade 8)
8.G.B.6Explain a proof of the Pythagorean Theorem and its converse (Using the converse of the Pythagorean theorem to conclude that a triangle with sides in ratio 3:4:5 is right-angled, with the 5-side as hypotenuse.)7.G.B.4Know the formulas for the area and circumference of a circle and use them to solve problems (Relating radius to diameter (diameter = 2r = 6) and identifying the hypotenuse of the inscribed right triangle as a diameter of the circle.)7.G.A.1Solve problems involving scale drawings of geometric figures, including computing actual lengths from a scale factor (Setting a scale factor k so the ratio sides 3,4,5 become real lengths, solving 5k = 6 for k = 1.2, and finding the legs 3.6 and 4.8.)6.G.A.1Find the area of right triangles, other triangles, and polygons (Computing the triangle's area as half the product of its two perpendicular legs, 1/2 (3.6)(4.8) = 8.64.)
⭐ A 3-4-5 triangle is a right triangle, so its longest side stretches across the circle as the diameter; scale everything to match, then take half of leg times leg.
⭐ A 3-4-5 triangle is a right triangle, so its longest side stretches across the circle as the diameter; scale everything to match, then take half of leg times leg.
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