Similar Triangles (Hard)

Late AMC 10/12 through early AIME — dense computation.

Similar Triangles: core idea

Nested figures.

In $\triangle ABC$ , $DE\parallel BC$ , $AD=6,DB=9,AE=8$ . Find $EC$ .

Hint: Match corresponding lengths; areas and volumes use powers of the scale. Answer: 12.

1. The altitude to a right triangle's hypotenuse splits it into $4,9$ . Find the altitude.
2. A $12\times35$ rectangle is similar to one with diagonal $74$ . Find its short side.
3. Similar triangles have area ratio $49:81$ and perimeter difference $16$ . Find the smaller perimeter.
4. A parallel cut leaves area $36$ from a triangle of area $100$ and base $25$ . Find the small base.
5. A $5$ - $12$ - $13$ triangle is enlarged to hypotenuse $39$ . Find its area.
6. A $6$ -foot person, $10$ feet from a lamp, casts a $5$ -foot shadow. Find lamp height.
7. A right triangle has hypotenuse $25$ and a leg projection $9$ . Find that leg.
8. Similar cones have heights $8,12$ ; the smaller volume is $216\pi$ . Find larger volume $/\pi$ .
9. A $7$ - $24$ - $25$ triangle is enlarged until area is $1176$ . Find its new hypotenuse.