# ISEE Upper Level Quantitative : Geometry

## Example Questions

### Example Question #21 : Right Triangles

is a right angle.

Which is the greater quantity?

(a)

(b)

(a) and (b) are equal

(a) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(b) is the greater quantity

(a) is the greater quantity

Explanation:

Corresponding angles of similar triangles are congruent, so, since , and  is right, it follows that

is a right angle of a right triangle . The other two angles must be acute - that is, with measure less than  -  so .

### Example Question #22 : Right Triangles

is inscribed in a circle.  is a right angle, and

Which is the greater quantity?

(a)

(b)

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

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

(a) and (b) are equal

Explanation:

The figure referenced is below:

has measure , so its corresponding minor arc, , has measure . The inscribed angle that intercepts this arc, which is , has measure half this, or . Since  is a right angle, the other acute angle, , has measure

Therefore, .

### Example Question #31 : Right Triangles

Consider a triangle, , in which , and . Which is the greater number?

(a) The measure of  in degrees

(b)

(b) is the greater quantity

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

(a) is the greater quantity

(a) and (b) are equal

(a) and (b) are equal

Explanation:

By the Converse of the Pythagorean Theorem, a triangle is right if and only if the sum of the squares of the lengths of the smallest two sides is equal to the square of the longest side. Compare the quantities  and

, so  is right, with the right angle opposite longest side . Thus,  is right and has degree measure 90.

### Example Question #61 : Geometry

The length of a side of a square is two-thirds the length of a leg of an isosceles right triangle. Which is the greater quantity?

(a) The area of the square

(b) The area of the triangle

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

(b) is greater

(b) is greater

Explanation:

Let  be the length of a leg of the right triangle. Then the sidelength of the square is  .

(a) The square has area

(b) The isosceles right triangle has base and height  area

, so (b) is greater.

### Example Question #65 : Geometry

Two triangles are on the coordinate plane. Each has a vertex at the origin.

Triangle A has its other two vertices at  and .

Triangle B has its other two vertices at   and .

Which is the greater quantity?

(a) The area of Triangle A

(b) The area of Triangle B

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

(a) and (b) are equal

Explanation:

Each triangle is a right triangle with legs along the - and -axes, so the area of each can be calculated by taking one-half the product of the two legs.

(a) The horizontal and vertical legs have measures 18 and , respectively, so the triangle has area .

(b) The horizontal and vertical legs have measures  and 9, respectively, so the triangle has area .

The areas are equal.

### Example Question #66 : Geometry

Construct rectangle , and locate midpoint  of side . Now construct segment .

Which is the greater quantity?

(b) Three times the area of

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

(a) is greater

(a) and (b) are equal

Explanation:

is a right triangle with right angle , so its legs measure  and ; its area is . Since  is the midpoint of , making the area of the triangle

Rectangle  has area

Quadrilateral ,  which is the portion of  not in , has as its area

Therefore, the area of Quadrilateral  is three times that of , making (a) and (b) equal.

### Example Question #67 : Geometry

Construct rectangle . Let  and  be the midpoints of  and , respectively, and draw the segments  and . Which is the greater quantity?

(a) The area of

(b)  The area of

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

(a) is greater.

(a) and (b) are equal.

Explanation:

Each triangle is a right triangle, and each has its two legs as its base and height.

(a)  is the midpoint of , so .

The area of  is .

(b)  is the midpoint of , so .

The area of  is

.

The triangles have equal area.

### Example Question #68 : Geometry

The length of a side of a square is one-half the length of the hypotenuse of a  triangle. Which is the greater quantity?

(a) The area of the square

(b) The area of the triangle

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

(a) is greater.

Explanation:

(a) Let  be the sidelength of the square. Then its area is .

(b) In a  triangle, the shorter leg is one-half as long as the hypotenuse. The hypotenuse has length , so the shorter leg has length . The longer leg is  times as long as the shorter leg, so the longer leg will have length . The area of the triangle is

.

, so ;  the square has the greater area.

### Example Question #69 : Geometry

Give the area of the above right triangle in terms of .

Explanation:

The area of a triangle is half the product of its base and its height; for a right triangle, the legs, which are perpendicular, serve as base and height.

### Example Question #70 : Geometry

Note: Figures NOT drawn to scale.

Refer to the above figures - a right triangle and a square. The area of the triangle is what percent of the area of the square?

Explanation:

The area of the triangle is

square inches.

The sidelength of the square is  inches, so the area of the square is

.

The question becomes "what percent of 576 is 270", which is answered as follows: