ISEE Upper Level Quantitative : How to find the area of a triangle

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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Example Questions

Example Question #1 : How To Find The Area Of A Triangle

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.

It is impossible to tell from the information given.

(b) is greater.

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 #2 : How To Find The Area Of A Triangle

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

It is impossible to tell from the information given

(b) is greater

(a) is greater

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 #3 : How To Find The Area Of A Triangle

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 #4 : How To Find The Area Of A Triangle

Pentagon 2

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

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

(b) is the greater quantity

(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.

Pentagon 3

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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