SAT and ACT Math Strategy: The “Three Right Triangles”
Making sense of “Three Right Triangle” questions (figure below) can be difficult. First, while there are two obvious right triangles (ΔADB and ΔBDC), the third is ‘lying’ on it’s hypotenuse! See the right angle at the top of the figure? That’s ΔABC.
In order to make sense of the relationship between these triangles, let’s first look at their angles:
To understand the relationship of the angles in this figure, let’s look at the largest triangle, ΔABC. It has a 90º angle, and two angles I’ve labeled aº and cº. Because this is a right triangle, a+c+90=180, or a+c=90. If that is the case, then any right triangle with either acute angle must have the other, as they sum to 90º. Let’s now update our drawing, applying this to the smaller two triangles (ΔADB and ΔBDC).
Now we’ve clearly proven all three triangles are similar: they share the same angles and thus have proportional side-lengths. But applying this information can still be tricky with the current figure, so let’s extract each triangle from the figure, drawing them in a row:
Hmm, this is certainly more clear, but it’s not very helpful that each triangle is lying on a different side. For simplicity’s sake, let’s call the side adjacent to angle aº the base and the side adjacent to angle cº the height. We should now be able to put each of these triangles “right side down.” See for yourself below:
Let’s now apply all this to a real-life example from the ACT:
In order to find line AD, first, label your angles (I like to use the same letter as the closest vertex):
Remember our earlier proof that aº+cº=90? Well, let’s use that now, labeling the other angles:
Like before, think of the line adjacent to aº as the base and the line adjacent to cº as the height. Draw all three triangles separately, keeping track of all vertices, side lengths and angles:
Now that we’ve organized the given information in an easy-to-use format, solving the question will be much simpler. First, use the Pythagorean Theorem (or your knowledge of the Pythagorean triplets), to find the hypotenuse AC.
Now that we know AC=25, either use the ratio between triangles or a proportion (shown below), to solve. I’ve labeled the side length we want to find, AD, as x.
So, with organization, patience, the Pythagorean Theorem and proportions, the “three right triangles” question is no match for our geometry!