### All SAT Math Resources

## Example Questions

### Example Question #31 : Acute / Obtuse Triangles

An isosceles triangle has an area of 12. If the ratio of the base to the height is 3:2, what is the length of the two equal sides?

**Possible Answers:**

5

4

6

4√3

3√3

**Correct answer:**

5

Area of a triangle is ½ x base x height. Since base:height = 3:2, base = 1.5 height. Area = 12 = ½ x 1.5 height x height or 24/1.5 = height^{2}. Height = 4. Base = 1.5 height = 6. Half the base and the height form the legs of a right triangle, with an equal leg of the isosceles triangle as the hypotenuse. This is a 3-4-5 right triangle.

### Example Question #12 : Isosceles Triangles

Two sides of a triangle each have length 6. All of the following could be the length of the third side EXCEPT

**Possible Answers:**

**Correct answer:**12

This question is about the Triangle Inequality, which states that in a triangle with two sides A and B, the third side must be greater than the absolute value of the difference between A and B and smaller than the sum of A and B.

Applying the Triangle Inequality to this problem, we see that the third side must be greater than the absolute value of the difference between the other two sides, which is |6-6|=0, and smaller than the sum of the two other sides, which is 6+6=12. The only answer choice that does not satisfy this range of possible values is 12 since the third side must be LESS than 12.

### Example Question #13 : Isosceles Triangles

A triangle has the following side lengths:

Which of the following correctly describes the triangle?

**Possible Answers:**

Obtuse and scalene

Obtuse and isosceles

Acute and scalene

None of these

Acute and isosceles

**Correct answer:**

Acute and isosceles

The triangle has two sides of equal length, 13, so it is by definition isosceles.

To determine whether the triangle is acute, right, or obtuse, compare the sum of the squares of the lengths of the two shortest sides to the square of the length of the longest side. The former quantity is equal to

The latter quantity is equal to

The former is greater than the latter; consequently, the triangle is acute. The correct response is that the triangle is acute and isosceles.