AMC 10 · 2005 · #6
Grade 8 geometry-2dPick an answer.
Tool #1 (Draw a Diagram) turns the words into a picture: an isosceles triangle standing on base AB with D marked farther along the line. The key move the picture reveals is dropping a perpendicular from the apex C straight down to the line, which splits everything into right triangles. Tool #7 (Identify Subproblems) then handles two right triangles in turn — one to find the height, one that contains CD. Tool #4 (Introduce a Variable) names BD so the second right triangle becomes an equation we can solve.
Drop a perpendicular from C
Symmetry puts the foot of the height at the base's midpoint.
An isosceles triangle folds onto itself along the line from its apex to the middle of the base, so that line cuts the base exactly in half.
The line from the apex to the middle of the base meets the base square on, splitting it exactly in half.
▸ Why?
Two equal sides make the triangle a mirror image of itself, so it folds onto itself along that line.
▸ Why?
A fold line meets what it folds at right angles and cuts it exactly in half.
Find the height CM
One right triangle gives the height.
The height and half the base are the two legs of a right triangle whose hypotenuse is the known slanted side.
8.G.B.7Identify SubproblemsSet up the right triangle with CD
A second right triangle brings in the given long distance.
The same height CM still stands over the line, so C, M, and D form another right triangle sharing that height.
8.G.B.7Introduce A VariableSolve for BD
Solving gives 3, choice (A).
Undo the squaring by taking a square root, keeping only the positive value because it measures a distance.
8.EE.A.2Introduce A VariableDrop a straight height from the tip of an isosceles triangle to split it into right triangles, then let the Pythagorean theorem turn the known lengths into an equation for the one you want.
- Drop a perpendicular from C
- Find the height CM
- Set up the right triangle with CD
- Solve for BD