# ISEE Upper Level Quantitative : Geometry

## Example Questions

### Example Question #21 : Triangles

is acute; . Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

Correct answer:

(b) is greater.

Explanation:

Since  is an acute triangle,  is an acute angle, and

,

(b) is the greater quantity.

### Example Question #2 : How To Find The Length Of The Side Of A Triangle

Given: . Which is the greater quantity?

(a) 18

(b)

Possible Answers:

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

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

Correct answer:

(a) is the greater quantity

Explanation:

Suppose there exists a second triangle  such that  and . Whether , the angle opposite the longest side, is acute, right, or obtuse can be determined by comparing the sum of the squares of the lengths of the shortest sides to the square of the length of the longest:

, making  obtuse, so .

We know that

and

.

Between  and , we have two sets of congruent sides, with the included angle of the latter of greater measure than that of the former. It follows from the Side-Angle-Side Inequality (or Hinge) Theorem that between the third sides,  is the longer. Therefore,

.

### Example Question #21 : Geometry

Which is the greater quantity?

(a)

(b)

Possible Answers:

(b) is greater

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

Correct answer:

(a) and (b) are equal

Explanation:

, so by definition, the sides are in proportion.

(a)

Substitute and solve for :

(b)

Substitute and solve for :

The two are equal.

### Example Question #21 : Geometry

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

Correct answer:

(a) is greater.

Explanation:

, so by definition, the sides are in proportion. Therefore,

.

Substitute:

, so (a) is greater.

### Example Question #22 : 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:

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

Correct answer:

(a) and (b) are equal.

Explanation:

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 #21 : 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:

(a) and (b) are equal

(b) is greater

(a) is greater

It is impossible to tell from the information given

Correct answer:

(b) is greater

Explanation:

(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 #22 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

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

Explanation:

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 #1 : How To Find The Area Of A Triangle

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:

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

Correct answer:

(a) and (b) are equal

Explanation:

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.

### Example Question #1 : Equilateral Triangles

is an equilateral triangle. Points  are the midpoints of , respectively.  is constructed.

Which is the greater quantity?

(a) The perimeter of

(b) Twice the perimeter of

Possible Answers:

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

If segments are constructed in which the endpoints form the midpoints of the sides of a triangle, then each of the sides of the smaller triangle is half as long as the side of the larger triangle that it does not touch. Therefore:

The perimeter of  is:

,

which is twice the perimeter of .

Note that the fact that the triangle is equilateral is irrelevant.

### Example Question #1 : Equilateral Triangles

Column A                 Column B

The perimeter           The perimeter

of a square with        of an equilateral

sides of 4 cm.           triangle with a side

of 9 cm.

Possible Answers:

The quantity in Column A is greater.

There is not enough info to determine a relationship between the columns.

The quantities in both columns are equal.

The quantity in Column B is greater.

Correct answer:

The quantity in Column B is greater.

Explanation:

Perimeter involves adding up all of the sides of the shape. Therefore, the square's perimeter is or 16. An equialteral shape means that all of the sides are equal. Therefore, the perimeter of the triangle is or 27. Therefore, Column B is greater.