AMC 10 · 2017 · #24
Grade 11 geometry-2dPick an answer.
Neither XF nor XG is easy to find on its own, but the question asks only for their product. Split the work into three independent pieces. The two parallel lines force a chain of similar triangles that expresses XF as a fixed multiple of XC. Power of a Point converts XC · XG into XB · XD, which lives entirely on the diagonal. The only genuine measurement left is BD, and the Law of Cosines pulls that out of the four side lengths. When the pieces are multiplied, XC cancels, so no length along line CX ever has to be computed.
Draw it and name one length
One letter names the whole diagonal.
One diagonal carries all four special points, so naming its length turns the whole picture into fractions of a single number.
10.G-CO.A.1Draw A DiagramMeasure the gap XY
The given fractions place both marked points.
Both points are pinned as fractions of the same segment, so the gap between them is just a subtraction of fractions.
7.RP.A.3Introduce A VariableParallel to AD gives the first similar pair
The first parallel gives a ratio.
Parallel lines copy angles across a transversal, and the crossing point supplies the vertical angle, which is exactly AA.
10.G-SRT.A.2Identify SubproblemsParallel to AC passes the ratio along
The second passes that ratio along.
A line parallel to AC cutting two rays out of X stretches both rays by the very same factor.
10.G-SRT.B.5Identify SubproblemsPower of a Point handles XG
The power of a point handles the circle.
Every chord through a fixed interior point splits into two pieces whose product is the same, no matter which chord you pick.
Every chord through a fixed interior point splits into two pieces whose product is always the same.
▸ Why?
The two chords cut out triangles with matching angles, so their sides sit in one fixed ratio.
▸ Why?
Every point of the circle is one radius from the centre, which is what ties all the chords to one number.
Multiply so that XC cancels
Multiplying makes one length cancel.
Grouping a product the right way lets an unknown factor cancel instead of forcing you to hunt it down.
9.A-SSE.A.2Change Focus Count The ComplementWrite BD squared two ways
Two expressions for the diagonal agree.
One diagonal belongs to two triangles at once, and the cyclic condition ties their angles together, so the two expressions for it must agree.
11.G-SRT.D.11Introduce A VariableSolve for the cosine and finish
That gives 17, choice (A).
Naming cos∠ BAD as the unknown turns a two-sided geometric statement into one linear equation.
9.A-REI.B.3Introduce A VariableWhen a problem asks for a product, look for a way to make the unknown factor cancel instead of chasing it: here every length along line CX disappears and only the diagonal BD is left to find.
- Draw it and name one length
- Measure the gap XY
- Parallel to AD gives the first similar pair
- Parallel to AC passes the ratio along
- Power of a Point handles XG
- Multiply so that XC cancels
- Write BD squared two ways
- Solve for the cosine and finish
More like this
Same archetype — closest grade level first.
