How to Solve Inscribed Triangles

A common geometry problem is one where you're given a right triangle with a line going through the middle and you're asked to find one or more missing side lengths.

These questions can often be paralyzing - not because you don't have the skills to solve it, but because there's so much going on that it's hard to know how to set it up. Here's a step-by-step guide for how to solve inscribed triangles.

Step 1: label everything. Assign letters, tick marks, colors, or symbols to each of the unknown sides and angles to help you keep track of what's what, because you'll need to use a lot of them along the way.

Step 2: redraw the triangles separately. You should have three triangles total: the big outer one, and the two smaller inner ones. The goal is to redraw them to all be facing the same direction, but it can also be helpful to draw them separately in their original orientation first before rotating them.

When you rotate them, make sure to pay attention to the angles to help you figure out which sides go where. In this example, I'm rotating the two smaller triangles to match the orientation of the large triangle, so the angles marked with a single blue tick mark are on the bottom left, and the angles marked with double red tick marks are on the top right. This helps me identify where the side lengths go - I know that side "a" has to touch the blue angle and side "b" has to touch the red angle.

Step 3: Set up the Pythagorean Theorem equation for each of the three triangles, and solve if possible. In most cases, only one of the triangles will have enough information to be solvable at this point.

'Step 4: Set up ratios of similar triangles. (TIP: Watch out to make sure you're setting them up correctly. Whichever piece is on top in one fraction should have its corresponding piece on top in the second fraction. Also, make sure to take into account the value you found on step 3!)

Step 5: Look for an equation that can be solved to find the specific value the problem is asking for. If there isn't one yet, then find which equations are solvable, solve them, and plug the values in.

Step 6: Solve the equation for the variable we want! 

(Make sure to put your final answer in whatever format is specified by the assignment or your teacher. Don't forget to reduce fractions where necessary!)