Competition · AMC preparation · step 4 of 4
AMC 8 · 2007 · #14
Grade 8 geometry-2dPick an answer.
AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #1 (Draw a Diagram) is the natural opening: sketch the isosceles triangle and drop the altitude from the apex to the base. Because the triangle is isosceles, that altitude bisects the base, splitting the figure into two congruent right triangles. The picture turns the problem into a Pythagorean Theorem exercise where the congruent side is the hypotenuse. Tool #5 (Look for a Pattern) is the finisher: once the two legs are 5 and 12, recognizing the classic 5-12-13 Pythagorean triple gives the hypotenuse instantly — no square root needed.
Sketch the triangle and midpoint
Drop the altitude from apex A to base BC; in an isosceles triangle it bisects the base, so BM = MC = 12 and a right angle appears at M.
Adding the altitude is the Grade 4 move of drawing a perpendicular line to expose a right angle hidden in the figure.
4.G.A.1Draw A DiagramFind the altitude from the area
Read the area formula backwards: 60 = × 24 × AM forces the altitude AM = 5.
The Grade 6 triangle-area formula reads backwards: given area and base, the height is forced.
6.G.A.1Draw A DiagramUse the right triangle
△ AMB has legs AM = 5 and BM = 12, so hypotenuse AB is the 5-12-13 triple's 13 — no square root needed.
Recognizing the 5-12-13 triple is a Grade 8 Pythagorean Theorem shortcut. The pattern saves the square root.
The congruent side is the hypotenuse of a right triangle whose legs are 5 and 12, so its length is completely fixed.
▸ Why?
Dropping the altitude from the apex splits the isosceles triangle into two identical right triangles, and in each one the congruent side lies opposite the right angle, making it the hypotenuse while the other leg is half the base, 12.
▸ Why?
The triangle has two equal sides, so flipping it across the altitude lays one half exactly onto the other; the matched halves share the same right angle at the foot and split the base into two equal parts of 12.
▸ Why?
Once a right triangle's two legs are fixed at 5 and 12, only one length can close the triangle as the hypotenuse, so the congruent side is forced.
▸ Why?
In a right triangle the two legs' squares add up to the hypotenuse's square, so 5 and 12 leave exactly one hypotenuse: 5²+12²=25+144=169, whose square gives the third side.
Drop the altitude in an isosceles triangle and it bisects the base — the picture hands you a right triangle, and the 5-12-13 Pythagorean triple finishes the problem.
- Sketch the triangle and midpoint
- Find the altitude from the area
- Use the right triangle
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