# ISEE Upper Level Quantitative : Triangles

## Example Questions

### Example Question #21 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

is acute; . Which is the greater quantity?

(a)

(b)

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

(b) is greater.

Explanation:

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

,

(b) is the greater quantity.

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

Given: . Which is the greater quantity?

(a) 18

(b)

(a) and (b) are equal

(a) is the greater quantity

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

(b) is the greater quantity

(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 #1 : How To Find If Two Triangles Are Similar

Which is the greater quantity?

(a)

(b)

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:

, so by definition, the sides are in proportion.

(a)

Substitute and solve for :

(b)

Substitute and solve for :

The two are equal.

### Example Question #2 : How To Find If Two Triangles Are Similar

Which is the greater quantity?

(a)

(b)

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

(a) and (b) are equal.

(a) is greater.

Explanation:

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

.

Substitute:

, so (a) is greater.

### Example Question #23 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

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

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

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

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

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given

(b) is greater

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

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

None of the other responses is correct

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

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

(a) The area of

(b) The area of

(a) is the greater quantity

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

(a) and (b) are equal

(b) is the greater quantity

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

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

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

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

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.

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

The quantity in Column A is greater.

The quantities in both columns are equal.

The quantity in Column B is greater.