Similar triangles on ACT Math

Two triangles are similar when they have the same shape but not necessarily the same size, which means their corresponding angles are equal and their corresponding sides are proportional. Once you spot similarity, the work is almost always the same: write a proportion, cross multiply, solve. The hard part is not the algebra, it is making sure the sides you pair up actually correspond.

How to know two triangles are similar

You do not need all three angles. If two angles of one triangle match two angles of another, the third pair has to match as well, since all three add to 180 degrees. That is the test the ACT leans on, and it usually appears in disguise: shared angles, vertical angles, or angles formed by parallel lines. Similar is not the same as congruent, though. Congruent triangles have equal sides; similar triangles are scaled copies.

Set up the proportion so the parts match

This is where most points are lost. Write your ratio in a consistent order, small triangle on top and large triangle on the bottom, and keep that order for both fractions. Say the shorter sides of two similar triangles are 6 and 9, and the side of the small triangle that corresponds to an unknown x measures 8. Then 6 over 9 equals 8 over x. Cross multiplying gives 6 times x equals 72, so x is 12.

A sanity check catches most setup errors. The large triangle is 1.5 times the small one, and 12 is 1.5 times 8. If a side of the bigger triangle comes out smaller than its partner, you flipped the ratio.

The parallel line inside a triangle

Here is the setup that shows up more than any other. Triangle ABC has a segment DE drawn across it with D on side AB, E on side AC, and DE parallel to BC. Because of that parallel, angle ADE equals angle ABC and angle AED equals angle ACB, so the small triangle ADE is similar to the whole triangle ABC.

Suppose AD is 4, DB is 6, and AE is 6. What is EC? The clean way is to compare the pieces of each side: AD over DB equals AE over EC, so 4 over 6 equals 6 over EC. Cross multiplying gives 4 times EC equals 36, so EC is 9. Comparing whole sides works too: AB is 10, so 4 over 10 equals 6 over AC, which makes AC 15 and EC 9.

Shadows, ladders, and other word problems

When a problem mentions a shadow, a mirror on the ground, or two vertical objects in the same sunlight, it is a similar triangles problem wearing a costume. Each object and its shadow form the legs of a right triangle, and the rays hit both objects at the same angle.

A person 6 feet tall casts an 8 foot shadow at the same moment a flagpole casts a 40 foot shadow. How tall is the flagpole? Set height over shadow equal for both: 6 over 8 equals h over 40. Cross multiplying gives 8 times h equals 240, so h is 30 feet. Notice that both fractions were written the same way, height on top. Flipping one of them gives 6 over 8 equals 40 over h, which lands on about 53.3 instead.

Lengths scale by k, areas scale by k squared

If every length in a triangle is multiplied by a scale factor k, the area is multiplied by k squared. That follows from the area formula: both the base and the height pick up a factor of k.

Say two similar triangles have corresponding sides of 4 and 6, so k is 6 divided by 4, which is 1.5. Then the areas are in a ratio of 1.5 squared, which is 2.25. If the smaller triangle has an area of 20, the larger one has an area of 45. The trap answer, 30, comes from scaling the area by 1.5 instead of 2.25.

Traps to watch for

• Pairing sides by how they look in the figure rather than by which angles they sit across from. The two triangles are often drawn at different orientations on purpose.
• Confusing a piece of a side with the whole side, the usual slip in the parallel line setup. Decide up front whether you are comparing AD to AB or AD to DB, and label it.
• Assuming similarity from appearance alone. You need matching angles, parallel marks, or proportional sides stated in the problem.

A quick self-check

Two similar triangles have corresponding sides of 5 and 15. What is the scale factor, and if the smaller has an area of 8, what is the area of the larger? Then: in triangle ABC with DE parallel to BC, AD is 3, AB is 12, and AC is 20. Find AE. The answers are a scale factor of 3 with an area of 72, since 8 times 9 is 72, and AE equals 5, since 3 over 12 equals AE over 20. Drilling a batch of these on thirty-six until the proportion writes itself is the fastest way to lock this in. When you are ready, keep going with general ACT triangle problems and see where this fits in how much geometry is on the ACT.

This article offers general ACT prep guidance. The ACT can change from year to year, including its format, scoring, policies, test dates, and fees, so always confirm the latest details on the official ACT website at act.org before you make decisions. ACT® is a registered trademark of ACT, Inc. thirty-six is not affiliated with, endorsed by, or sponsored by ACT.