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EXTRA

Unit 32: Triangles

There are three common types of triangle problems on the SAT.  They mostly concern:

  • Similar triangles
  • Special right triangles
  • Pythagorean theorem and triplets

Similar Triangles

Similar triangles have three notable things about them.  They have the same angles, the same ratio of sides, and the same sine, cosine, and tangent.  

Note that congruent triangles are just a specific type of similar triangle – their angles are the same, and the ratio of the sides is 1:1 (because they are exactly the same triangle).

The SAT tries to trick you by making them hard to recognize as similar triangles.  Some examples:  

**Note that in all these examples, there are two triangles that share the same angles, and thus have the same ratio of sides.  Anytime you see two triangles, or one triangle with any kind of line through it, check to see if you have similar triangles!

Strategy:

  1. Mark the sides of the similar triangles that correspond to one another.
  2. Create a ratio with a side of the smaller triangle over the corresponding side from the larger triangle.
  3. Create ratios for the other corresponding sides, set the ratios all equal, and cross multiply to solve.

Special Right Triangles

There are two types of special right triangles you’ll see on the SAT.  When you see a right triangle (a triangle that contains a 90-degree angle, denoted by the little  in the corner), you should always check to see if it’s a right triangle.  The two types are:

1. 30°-60°-90°

2. 45°-45°-90°

What makes the first two types of triangles “special” is that if you know the angles and one of the side lengths, you can figure out the other two sides. 

Fortunately for us, the SAT gives you the first two in the reference section at the front of each math section!  It looks like this:

If you have a 30°-60°-90°, and you are given that the side opposite the 30° angle is 10, then you know the hypotenuse is 20 and the side opposite the 60° angle is(10sqrt3).  

Pythagorean Theorem and Triplets

Remember that the Pythagorean theorem can only be used with right triangles, and it is:

$$a^2+b^2=c^2$$

There are two Pythagorean triplets that can save you time on the SAT. They’re common side lengths that go together in a right triangle.

1. 3-4-5

2. 5-12-13

So given a triangle with side lengths of 

We could plug in  (a^2+8^2=10^2) to then get  (a^2+64=100) and solve to get a

OR

We can look and see one side is a multiple of 5 (10), one is a multiple of 4 (8), so the third side MUST be a multiple of 3 (6).  This can save you a decent amount of time and prevent calculation errors, so keep an eye out for the 3-4-5 triangles as well.

Traps

Hidden special right triangles
Always be on the lookout for special right triangles. The SAT loves to include special right triangles to test if a student is looking for them. They’re usually the key to solving the problem.

Similar triangles in unsimilar positions
Sometimes two triangles will be similar, but one will be positioned on its side, while the other is standing up, or they’ll be facing opposite direction. Take the time to ensure you’ve matched the correct corresponding sides.
 
Redrawing the triangles yourself can often help with this one.