Similar Triangles (AIME / USAJMO / AMO)

Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.

Proof track: treat each problem as a full olympiad writeup on paper. Here you only enter the final answer (integer or short expression).

Easy Medium Hard AIME / USAJMO / AMO

Similar Triangles: core idea

Angle chase proofs.

Example problem: hint and answer

**(Proof track.)** Give a complete justification. In $\triangle ABC$ , $D\in AB,E\in AC,DE\parallel BC$ , $AD:DB=2:3$ , and $[ADE]=32$ . Find $[ABC]$ .

Hint: Establish the similarity correspondence before using a scale ratio. Answer: 200.

Practice (9 problems — type your answers)

No proof submission required — answers only.

1. **(Proof track.)** Give a complete justification. A right triangle's altitude creates hypotenuse segments $5,20$ . Prove the leg lengths relation and find the shorter leg.
2. **(Proof track.)** Give a complete justification. Two similar triangles have circumradii $9,15$ and area difference $128$ . Find the smaller area.
3. **(Proof track.)** Give a complete justification. A $4\times9$ photograph is enlarged so its diagonal is $13\sqrt{97}$ . Find its area.
4. **(Proof track.)** Give a complete justification. In a triangle, a parallel segment cuts off perimeter $18$ from a triangle of perimeter $45$ . If the large area is $200$ , find the small area.
5. **(Proof track.)** Give a complete justification. A $9$ -foot pole casts $12$ feet of shadow. At that instant, a building's shadow is $80$ feet. Find its height.
6. **(Proof track.)** Give a complete justification. Similar triangles have side ratio $3:5$ and their corresponding altitudes differ by $14$ . Find the larger altitude.
7. **(Proof track.)** Give a complete justification. A right triangle has legs $15,20$ . Its altitude to hypotenuse cuts a smaller triangle. Find that smaller triangle's area.
8. **(Proof track.)** Give a complete justification. A triangle similar to one with sides $8,15,17$ has perimeter $120$ . Find its area.
9. **(Proof track.)** Give a complete justification. A line parallel to one side divides a triangle into a top triangle and trapezoid of equal area. Find the top-to-whole side ratio.