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2011 AMC 8

Problem 16

Problem 16 · 2011 AMC 8 Medium
Geometry & Measurement isosceles-altitudepythagorean-triple

Let A be the area of the triangle with sides of length 25, 25, and 30. Let B be the area of the triangle with sides of length 25, 25, and 40. What is the relationship between A and B?

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Answer: C — A = B.
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Hint 1 of 2
Both triangles are isosceles with the same equal sides (25). Drop the altitude to the odd side — it splits each into two right triangles whose hypotenuse is 25.
Still stuck? Show hint 2 →
Hint 2 of 2
Watch what the Pythagorean theorem produces: both right triangles are the 15-20-25 triangle (a 3-4-5 scaled by 5). They're the same triangle — the 15 and 20 just swap between ‘half-base’ and ‘height.’
Show solution
Approach: drop altitudes — both triangles are built from the identical 15-20-25 right triangle
  1. Base 30 (half-base 15, slant 25): height = √(252 − 152) = 20, so A = (1/2)(30)(20) = 300.
  2. Base 40 (half-base 20, slant 25): height = √(252 − 202) = 15, so B = (1/2)(40)(15) = 300.
  3. A = B.
  4. The deep reason: each triangle is two copies of the same 15-20-25 right triangle. In one, the ‘15’ leg is the half-base and ‘20’ is the height; in the other they trade places. Since area only uses the product of base-half and height, swapping them changes nothing — so the areas had to match.
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