AMC 10 · 2008 · #10
Grade 8 geometry-2dPick an answer.
AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The problem is pure circle geometry, so tool #1 (Draw a Diagram) leads: sketching the center, the chord, and the arc midpoint exposes a line of symmetry that lines up the center O, the chord's midpoint M, and C. Once that line is drawn, the distance AC breaks into small right-triangle pieces, so tool #7 (Identify Subproblems) handles them in order — first the distance from the center to the chord, then the short gap MC, then AC itself. Tool #3 (Eliminate Possibilities) closes it out: the answer sits just above 3, and only one choice is that small.
Draw the axis of symmetry
Let O be the center and M the midpoint of AB. The mirror line through O and M also passes through C and meets AB squarely, so AM=MB=3.
Reflecting the circle across the diameter through the arc's midpoint swaps A and B, so that diameter must pass through C and cut the chord in half.
8.G.A.1Draw A DiagramDistance from center to chord
Triangle OAM is right-angled at M, with radius OA=5 and AM=3, so OM²=25-9=16 and OM=4.
The line from the center to a chord's midpoint is perpendicular to the chord, so half the chord and the center-to-chord distance are the two legs of a right triangle.
The line from the centre to a chord's midpoint meets the chord square on.
▸ Why?
The centre is equally far from both ends of the chord, so that line is the fold that halves it.
▸ Why?
That right angle ties the half-chord, the centre distance, and the radius into one equation.
Find the short gap MC
On that line M sits at 4 and C, being on the circle, sits at the radius 5, so the gap is MC=5-4=1.
O, M, and C sit on one line with M between O and C, so the distances simply subtract.
6.NS.C.6Identify SubproblemsCompute AC
Triangle AMC is right-angled at M with legs AM=3 and MC=1, so AC²=9+1=10 and AC=√(10), choice (A).
AC is the hypotenuse of a small right triangle whose legs are the half-chord AM and the short reach MC.
8.G.B.7Identify SubproblemsDraw the line of symmetry through the center: it splits the chord in half and turns the distance you want into the hypotenuse of a tiny right triangle.
- Draw the axis of symmetry
- Distance from center to chord
- Find the short gap MC
- Compute AC