The third side of a triangle is always greater than the difference of the other two sides and less than the sum of the other two sides. This applies to every side of a triangle. In other words, you can arbitrarily pick any one side to be the “third side,” and then that side must be greater than the difference of the other two and less than the sum of those two.

Here that means that the third side must greater than the difference of  and . Since , that means that  is not an option. It also means that the third side must be less than the sum of  and . Since , that rules out  as an option. You know that the third side must be greater than  and less than : only , option II, fits.

Whether you use the Pythagorean Theorem or you quickly recognize that this is a  triangle, you can solve for the length of side .

You can prove that this is a  triangle because the hypotenuse is twice as long as one of the legs of the triangle. This then fits the side ratio , so you can fill in the middle side as .

Or you can use Pythagorean Theorem. Since you know side YZ is the hypotenuse, you can set it up as . This means that , so  and .

Once you've determined that , you can calculate the area. The area is half the base times the height, where the base and height form a -degree angle. This means that you can use  and . .

There is a rule about isosceles triangles that isn't obvious the first time you see it, but that the SAT likes to test: if an isosceles triangle includes a -degree angle, then it must be an equilateral triangle.

You can prove this by testing the cases: if you know that you have an isosceles triangle with a measure of  for one angle, then you can call your angles , , and . You know that  must sum to  (a rule of triangles), and that one of the following cases must be true:

 matches . if that's true, then your three angles are , , and . Since , that means that . Here, all three angles are .

 matches . The same as the above.

 matches . This then means that , so . If , then  so x = 60, which means that . Here, again, all angles are .

This problem blends two important rules related to triangles:

1) The sum of the interior angles of a triangle is .

2) In an isosceles triangle, two angles have the same measure.

Here, although you're not explicitly told which angles have the same measure, you can deduce that it must be angles XZY and YXZ - the two angles that do not measure  degrees. Note that if  were to be the "match," then you would already have  degrees assigned to those two angles, but that would violate the  rule.

Therefore, you know that your three angles are , , and  (where  represents the unknown, matching angle). You can then say that , so  and .

One of the most convenient things about isosceles right triangles is that you can use the two shorter sides as the base and the height to find the area, since they're connected by a right angle:

So if you know that  is the area in an isosceles right triangle, you can use  to solve for  as the length of each of the shorter sides. This means that , which you can simplify to:

And then solve for .

Because this is an isosceles right triangle, the sides will form the ratio , meaning that the hypotenuse will measure . If you sum the two shorter sides of  with the hypotenuse of , you reach .

This problem forces you to visualize a right triangle from the information provided in the story. When Mirinda makes a turn from heading directly south to heading directly west, that is a -degree right angle. And the shortest distance to get from her endpoint ( km south and  km west) back home is a diagonal line that connects the triangle:

So that shortest distance home is the hypotenuse of a right triangle with shorter sides  and .

To solve, you can use the Pythagorean Theorem: , where  and  are lengths of the shorter sides and  is the length of the hypotenuse. This then means that:

This simplifies to:

So  and .

The key to this problem is recognizing the two relationships here that must sum to  degrees: the three angles in a triangle must sum to  , and supplementary angles (those that are formed by splitting a straight line) must also sum to . Using those relationships, you should see that:

In the big triangle (JXZ), two angles are already given as  and , meaning that angle KZY must equal , based on the sum of angles in that triangle needing to be  .

Since that angle KZY is , that means that , as it has to sum with its supplementary angle to .

And using the small triangle to the right, KYZ, if you know that the two bottom angles (KYZ and KZY) are  and , then the third angle (YKZ at the top of that triangle) must account for the remaining .

And then since a is supplementary to that -degree angle, it has to be .

So  and , meaning that the sum is .

One triangle rule you must know is that the sum of a triangle's three angles must equal . Here that is important because you're given one angle  and part of another , and then asked to relate two variables together. You therefore know that  must equal . Using that equation:

Your job is to solve for  in terms of . So first combine like terms:

Then subtract  and  from both sides to isolate :

And you'll see that that is your answer.