AMC 10 · 2015 · #13

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length
coordinate-geometryarea-trianglesinteger-pythagorean-triples identify-subproblems ↑ Prerequisites: area-trianglescoordinate-geometry
📏 Medium solution 💡 3 insights
Problem
The line 12x+5y=60 cuts across the first quadrant and, together with the x-axis and the y-axis, encloses a triangle. Find the three altitudes of that triangle (one altitude from each vertex to the opposite side) and add their lengths.

Pick an answer.

(A)
20
(B)
$\dfrac{360}{17}$
(C)
$\dfrac{107}{5}$
(D)
$\dfrac{43}{2}$
(E)
$\dfrac{281}{13}$

AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The problem is about a line and the axes, so tool #1 (Draw a Diagram) comes first: plotting the intercepts pins down the three corners and reveals a right triangle resting against the axes. Once the shape is a right triangle with legs on the axes, the three altitudes split into easy pieces, so tool #7 (Identify Subproblems) handles them one at a time — the two legs are themselves altitudes, and the third altitude (to the slanted hypotenuse) comes from the area. Tool #3 (Eliminate Possibilities) is a safety net: only one answer choice has a denominator of 13, which is exactly what the hypotenuse altitude forces.

1STEP 1

Find the three corners

Set y=0 to get x=5, and x=0 to get y=12; with the axes' meeting point, the three corners are (0,0), (5,0), (0,12).

y=0→ x=5; x=0→ y=12; corners: (0,0),(5,0),(0,12)
2STEP 2

See the right triangle and its hypotenuse

The legs on the axes are 5 and 12 and meet at a right angle, so by the Pythagorean theorem the hypotenuse is √(5²+12²)=13.

√(5²+12²)=√(25+144)=√(169)=13
3STEP 3

Two altitudes are the legs themselves

Because the two legs are perpendicular, each is the altitude when the other is the base, so two of the altitudes are the legs 12 and 5.

h₁=12, h₂=5
4STEP 4

Third altitude from the area

The area is 1/2·5·12=30, and it also equals 1/2·13·h₃, so the altitude to the hypotenuse is h₃=60/13.

1/2·13· h₃=30 → h₃=60/13
5STEP 5

Add the three altitudes

Add the heights: 5+12+60/13=(221+60)/13=281/13; only choice (E) has denominator 13, so the answer is (E).

5+12+60/13=(221+60)/13=281/13
Answer
281/13
Check the size: the three altitudes are 12, 5, and 60/13≈4.6, so the sum is about 21.6. The fraction 281/13≈21.6 agrees. It also makes sense that the altitude to the long hypotenuse (≈4.6) is the shortest of the three, since the longest side needs the shortest height to keep the area at 30.
💡Key takeaway

In a right triangle the two legs are already altitudes; for the slanted side, use the fact that the area stays the same to find its matching height.

  • Find the three corners
  • See the right triangle and its hypotenuse
  • Two altitudes are the legs themselves
  • Third altitude from the area
  • Add the three altitudes

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