### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #41 : Plane Geometry

and are right triangles, with right angles , respectively. and .

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

**Correct answer:**

(a) is greater.

Each right triangle is a triangle, making each triangle isosceles by the Converse of the Isosceles Triangle Theorem.

Since and are the right triangles, the legs are , and the hypotenuses are .

By the Theorem, and .

, so and subsequently, .

### Example Question #45 : Geometry

Figure NOT drawn to scale.

In the above figure, is a right angle.

What is the ratio of the area of to that of ?

**Possible Answers:**

12 to 5

13 to 5

169 to 25

144 to 25

**Correct answer:**

144 to 25

The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller, similar triangles.

The similarity ratio of to can be found by determining the ratio of one pair of corresponding sides; we will use the short leg of each, and .

is also the long leg of , so its length can be found using the Pythagorean Theorem:

The similarity ratio is therefore

.

The ratio of the areas is the square of this ratio:

- that is, 144 to 25.

Certified Tutor