Similar Triangles (Medium)

AMC 10/12 mid (≈Q9–16) — multi-step.

Similar Triangles: core idea

Ratio chase.

A $6$ -foot person stands $15$ feet from a lamp and casts a $10$ -foot shadow. Find the lamp height.

Hint: Use the scale factor; areas scale by its square. Answer: 15.

1. The areas of similar triangles are $64$ and $225$ . If a side of the first is $12$ , find the matching side of the second.
2. In $\triangle ABC$ , $DE\parallel BC$ , $AD=4$ , $DB=6$ , and $AE=8$ . Find $EC$ .
3. A triangle has sides $9,12,15$ . A similar triangle has area $150$ . Find its longest side.
4. A right triangle has altitude to its hypotenuse splitting it into lengths $4$ and $9$ . Find the altitude.
5. A $10$ by $24$ rectangle and a similar rectangle differ in area by a factor of $9$ . Find the second perimeter.
6. A line parallel to the base cuts off a triangle with area $16$ from a triangle of area $100$ . If the small base is $8$ , find the large base.
7. Two similar cones have radii $3$ and $5$ . If the smaller volume is $108$ , find the larger volume.
8. A $12$ -meter building has a shadow of length $9$ . At the same time, a flagpole has shadow $15$ . Find its height.
9. Similar polygons have corresponding side lengths $x+2$ and $3x-6$ , and their area ratio is $4:9$ . Find $x$ .