AMC 10 · 2013 · #19
Grade 7 geometry-2dPick an answer.
The two side pieces are hidden, so name them with letters. A circle through B and X lets me turn the side lengths into one clean product using the Power of a Point idea. That product is a fixed number, so I factor it to list the possible values of BC, then use the triangle inequality to erase every value but one.
Draw the circle and mark radii
The circle's radius equals a known side.
Everything on a circle is exactly one radius from the center, so B and X are both 86 from A.
7.G.B.4Draw A DiagramName the two side pieces
The two pieces get names.
Giving the unknown pieces names lets the geometry clue become an equation.
6.EE.B.6Introduce A VariableUse Power of a Point at C
The power of the far corner gives a fixed product.
From one outside point, every secant line hits the circle with the same product of distances.
From one outside point, every line through the circle gives the same product of the two distances.
▸ Why?
The two lines 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 the two lines to one number.
Factor 2013 to list the options
Factoring lists only four options.
The product is locked at 2013, so BC has to be one of its factors.
4.OA.B.4Introduce A VariableErase impossible lengths
Size rules out all but 61, choice (D).
A side of a triangle can never reach or pass the sum of the other two sides.
7.G.A.2Eliminate PossibilitiesFrom one point outside a circle, every straight cut gives the same product of distances, so name the pieces, get one product, and factor it.
- Draw the circle and mark radii
- Name the two side pieces
- Use Power of a Point at C
- Factor 2013 to list the options
- Erase impossible lengths