AMC 8 · 2010 · #23
Grade 8 geometry-2dProblem
Semicircles POQ and ROS pass through the center O. What is the ratio of the combined areas of the two semicircles to the area of circle O?
Pick an answer.
AMC 8 2010 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 circle $O$ is centered at the origin and passes through the four points $P(-1,1)$, $Q(1,1)$, $R(-1,-1)$, and $S(1,-1)$. Two semicircles are drawn: one on chord $PQ$ as its diameter (passing through center $O$) and one on chord $RS$ as its diameter (also passing through $O$). What is the ratio of the combined area of the two semicircles to the area of circle $O$?
Givens: Circle $O$ has center at the origin and passes through $P(-1,1)$, $Q(1,1)$, $R(-1,-1)$, $S(1,-1)$; Semicircle $POQ$ has $PQ$ as its diameter and passes through $O$; Semicircle $ROS$ has $RS$ as its diameter and passes through $O$; Answer choices: (A) $\tfrac{\sqrt{2}}{4}$, (B) $\tfrac{1}{2}$, (C) $\tfrac{2}{\pi}$, (D) $\tfrac{2}{3}$, (E) $\tfrac{\sqrt{2}}{2}$
Unknowns: The ratio (combined area of the two semicircles) : (area of circle $O$)
Understand
Restated: A circle $O$ is centered at the origin and passes through the four points $P(-1,1)$, $Q(1,1)$, $R(-1,-1)$, and $S(1,-1)$. Two semicircles are drawn: one on chord $PQ$ as its diameter (passing through center $O$) and one on chord $RS$ as its diameter (also passing through $O$). What is the ratio of the combined area of the two semicircles to the area of circle $O$?
Givens: Circle $O$ has center at the origin and passes through $P(-1,1)$, $Q(1,1)$, $R(-1,-1)$, $S(1,-1)$; Semicircle $POQ$ has $PQ$ as its diameter and passes through $O$; Semicircle $ROS$ has $RS$ as its diameter and passes through $O$; Answer choices: (A) $\tfrac{\sqrt{2}}{4}$, (B) $\tfrac{1}{2}$, (C) $\tfrac{2}{\pi}$, (D) $\tfrac{2}{3}$, (E) $\tfrac{\sqrt{2}}{2}$
Plan
Primary tool: #11 Use Coordinates
Secondary: #7 Identify Subproblems
The figure is already drawn on the coordinate plane with the four corner points labeled, so Tool #11 (Use Coordinates) turns every length into a distance calculation. Tool #7 (Identify Subproblems) splits the question into two clean pieces — find the big circle's radius from $O$ to $P$, and find each semicircle's radius from the chord length $PQ$ — so the area ratio at the end is just $\dfrac{2 \cdot \tfrac{1}{2}\pi r_{\text{small}}^2}{\pi r_{\text{big}}^2}$.
Execute — Answer: B
8.G.B.8 Step 1 - Find the radius of circle $O$.
- Since $O = (0,0)$ and $P = (-1, 1)$ lies on the circle, the radius is the distance $OP$.
- Apply the distance formula (Pythagorean Theorem on the legs $1$ and $1$).
💡 Distance between two points on the coordinate plane via the Pythagorean Theorem is the Grade 8 distance formula.
7.G.B.4 Step 2 Compute the area of circle $O$ using $A = \pi r^2$ with $r = \sqrt{2}$.
💡 Applying $A = \pi r^2$ to a circle is the standard Grade 7 area-of-a-circle formula.
6.NS.C.8 Step 3 - Find the diameter of each small semicircle.
- The chord $PQ$ runs from $(-1, 1)$ to $(1, 1)$ — both points share $y = 1$, so the chord is horizontal and its length is just the difference of $x$-coordinates.
- The diameter equals $PQ$, so each semicircle's radius is half of that.
💡 Reading horizontal distance off the coordinate plane as a difference of $x$-coordinates is a Grade 6 coordinate-plane skill.
7.G.B.4 Step 4 - Compute the combined area of the two semicircles.
- Each has radius $1$, so each has area $\tfrac{1}{2}\pi (1)^2 = \tfrac{\pi}{2}$.
- Two of them together make a full unit circle's worth of area.
💡 Splitting the combined area into "two halves of a unit circle" is the Tool #7 subproblems move.
6.RP.A.1 Step 5 Take the ratio of the combined semicircle area to the area of circle $O$.
💡 Expressing one quantity as a fraction of another is the Grade 6 ratio concept.
8.G.B.8 Find the radius of circle $O$. Since $O = (0,0)$ and $P = (-1, 1)$ lies on the c 7.G.B.4 Compute the area of circle $O$ using $A = \pi r^2$ with $r = \sqrt{2}$. 6.NS.C.8 Find the diameter of each small semicircle. The chord $PQ$ runs from $(-1, 1)$ t 7.G.B.4 Compute the combined area of the two semicircles. Each has radius $1$, so each h 6.RP.A.1 Take the ratio of the combined semicircle area to the area of circle $O$. Review
Reasonableness: The big circle has radius $\sqrt{2} \approx 1.41$, so its area $2\pi$ is exactly twice the area of a unit circle. The two semicircles glue together (in area) into one unit circle of area $\pi$. So the ratio must be $\pi : 2\pi = 1 : 2$, matching answer (B). The other choices fail simple sanity checks: $\tfrac{2}{\pi} \approx 0.637$ would be a dimensional mismatch (no stray $\pi$ survives the ratio), and $\tfrac{\sqrt{2}}{4}, \tfrac{\sqrt{2}}{2}$ would require some length, not area, in the ratio.
Alternative: Tool #2 (Find a Pattern) via scaling: any circle is similar to any other circle, so areas scale by the square of the radius. The small semicircles have radius $1$ and the big circle has radius $\sqrt{2}$, so $\dfrac{r_{\text{small}}^2}{r_{\text{big}}^2} = \dfrac{1}{2}$. Two semicircles equal one full small circle, so the combined area equals one small circle. The ratio of (one small circle) : (big circle) is $\dfrac{1}{2}$ — answer (B), with no $\pi$ ever needing to be computed.
CCSS standards used (min grade 8)
6.NS.C.8Solve real-world and mathematical problems by graphing points in the coordinate plane (Reading the horizontal length $PQ = 2$ directly from the $x$-coordinates of $P(-1,1)$ and $Q(1,1)$.)6.RP.A.1Understand the concept of a ratio (Expressing the final answer as the ratio $\dfrac{\pi}{2\pi} = \dfrac{1}{2}$ of the combined semicircle area to the big-circle area.)7.G.B.4Know the formulas for the area and circumference of a circle and use them to solve problems (Applying $A = \pi r^2$ to get the big circle's area $2\pi$ and each small semicircle's area $\tfrac{\pi}{2}$.)8.G.B.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system (Computing $OP = \sqrt{1^2 + 1^2} = \sqrt{2}$ as the radius of circle $O$.)
⭐ This AMC 8 geometry problem only needs the Grade 8 distance formula and the Grade 7 circle-area formula that you already know!
⭐ This AMC 8 geometry problem only needs the Grade 8 distance formula and the Grade 7 circle-area formula that you already know!
More like this
Same archetype — closest grade level first.
