### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #25 : Plane Geometry

Triangle B has a height that is twice that of Triangle A and a base that is one-half that of Triangle A. Which is the greater quantity?

(a) The area of Triangle A

(b) The area of Triangle B

**Possible Answers:**

(a) and (b) are equal.

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

**Correct answer:**

(a) and (b) are equal.

Let and be the base and height of Triangle A. Then the base and height of Triangle B are and , respectively.

(a) The area of Triangle A is .

(b) The area of Triangle B is .

Therefore, (a) and (b) are equal.

### Example Question #26 : Plane Geometry

Two triangles on the coordinate plane have a vertex at the origin and a vertex at , where .

Triangle A has its third vertex at .

Triangle B has its third vertex at .

Which is the greater quantity?

(a) The area of Triangle A

(b) The area of Triangle B

**Possible Answers:**

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

(b) is greater

**Correct answer:**

(b) is greater

(a) Triangle A has as its base the horizontal segment connecting and , the length of which is 10. Its (vertical) altitude is the segment from to this horizontal segment, which is part of the -axis; its height is therefore the -coordinate of this point, or .

The area of Triangle A is therefore

(b) Triangle B has as its base the vertical segment connecting and , the length of which is 10. Its (horizontal) altitude is the segment from to this vertical segment, which is part of the -axis; its height is therefore the -coordinate of this point, or .

The area of Triangle B is therefore

, so . (b), the area of Triangle B, is greater.

### Example Question #27 : Plane Geometry

A triangle has sides 30, 40, and 80. Give its area.

**Possible Answers:**

None of the other responses is correct

**Correct answer:**

None of the other responses is correct

By the Triangle Inequality Theorem, the sum of the lengths of the two shorter sides of a triangle must exceed the length of its longest side. However,

;

Therefore, this triangle cannot exist, and the correct answer is "none of the other responses is correct".

### Example Question #28 : Plane Geometry

The above depicts Square ; , and are the midpoints of , , and , respectively. Which is the greater quantity?

(a) The area of

(b) The area of

**Possible Answers:**

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

It is impossible to determine which is greater from the information given

**Correct answer:**

(a) and (b) are equal

For the sake of simplicity, assume that the square has sidelength 2; this reasoning is independent of the actual sidelength.

Since , , and are the midpoints of their respective sides, , as shown in this diagram.

The area of , it being a right triangle, is half the product of the lengths of its legs:

The area of is half the product of the length of a base and the height. Using as the base, and as an altitude:

The two triangles have the same area.

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