Similar figures

Triangles ADE and ABC are similar. Their area ratio is the square of their length ratio. Find the smaller triangle area and subtract it from the whole triangle area.

First add AD and DB to find the whole side AB. Similar triangles ADE and ABC give DE / BC = AD / AB. DB is not the whole triangle side.

Find the full tree shadow length, rather than using only the difference. Heights and shadows form similar right triangles, so the height ratio equals the shadow length ratio.

Take the square root of the area ratio to find the length ratio. Use the vertex order to match BC to its corresponding side, then apply that ratio.

Find the perimeter ratio by taking the square root of the area ratio. Express both perimeters as parts of that ratio. Their difference gives the value of one part and then the smaller perimeter.

Triangles AOB and COD have equal opposite angles at O; the parallel bases give another equal angle pair. Match corresponding sides and form a proportion using the bases and the diagonal segments.

Check which facts are given: two equal angle pairs; two proportional side pairs and an equal included angle; or three proportional side pairs. Two proportional side pairs alone, without an angle, are insufficient.

The perimeter ratio equals the length ratio. Areas scale by the square of that ratio, rather than by the length factor alone.