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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:

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

Correct answer:

(a) and (b) are equal.

Explanation:

Let $\displaystyle b$ and $\displaystyle h$ be the base and height of Triangle A. Then the base and height of Triangle B are $\frac{1}{2} b$ $\displaystyle \frac{1}{2} b$ and $\displaystyle 2h$, respectively.

(a) The area of Triangle A is $A = \frac{1}{2} bh$ $\displaystyle A = \frac{1}{2} bh$.

(b) The area of Triangle B is $A = \frac{1}{2}\cdot \frac{1}{2} b \cdot 2h = \frac{1}{2} bh$ $\displaystyle A = \frac{1}{2}\cdot \frac{1}{2} b \cdot 2h = \frac{1}{2} bh$.

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

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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 (c,d) $\displaystyle (c,d)$, where c > d>0 $\displaystyle c > d>0$.

Triangle A has its third vertex at (10,0) $\displaystyle (10,0)$.

Triangle B has its third vertex at (0,10) $\displaystyle (0,10)$.

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

(b) is greater

(a) and (b) are equal

(a) is greater

Correct answer:

(b) is greater

Explanation:

(a) Triangle A has as its base the horizontal segment connecting (0,0) $\displaystyle (0,0)$ and (10,0) $\displaystyle (10,0)$, the length of which is 10. Its (vertical) altitude is the segment from (c,d) $\displaystyle (c,d)$ to this horizontal segment, which is part of the $\displaystyle x$-axis; its height is therefore the $\displaystyle y$-coordinate of this point, or $\displaystyle d$.

The area of Triangle A is therefore $A = \frac{1}{2} \cdot 10 \cdot d = 5d$ $\displaystyle A = \frac{1}{2} \cdot 10 \cdot d = 5d$

(b) Triangle B has as its base the vertical segment connecting (0,0) $\displaystyle (0,0)$ and (0,10) $\displaystyle (0,10)$, the length of which is 10. Its (horizontal) altitude is the segment from (c,d) $\displaystyle (c,d)$ to this vertical segment, which is part of the $\displaystyle y$-axis; its height is therefore the $\displaystyle x$-coordinate of this point, or $\displaystyle c$.

The area of Triangle B is therefore $A = \frac{1}{2} \cdot 10 \cdot c = 5c$ $\displaystyle A = \frac{1}{2} \cdot 10 \cdot c = 5c$

c > d $\displaystyle c > d$, so 5c > 5d $\displaystyle 5c > 5d$. (b), the area of Triangle B, is greater.

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

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

Possible Answers:

$75 \sqrt{105}$ $\displaystyle 75 \sqrt{105}$

600 $\displaystyle 600$

None of the other responses is correct

1,200 $\displaystyle 1,200$

1,600 $\displaystyle 1,600$

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,

$\displaystyle 30 + 40 = 70 < 80$;

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

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

Pentagon 2

The above depicts Square ABCD $\displaystyle ABCD$; X, Y $\displaystyle X, Y$, and $\displaystyle Z$ are the midpoints of $\overline{AD}$ $\displaystyle \overline{AD}$, $\overline{BC}$ $\displaystyle \overline{BC}$, and $\overline{AB}$ $\displaystyle \overline{AB}$, respectively. Which is the greater quantity?

(a) The area of $\bigtriangleup BZY$ $\displaystyle \bigtriangleup BZY$

(b) The area of $\bigtriangleup ZXD$ $\displaystyle \bigtriangleup ZXD$

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 $\displaystyle X$, $\displaystyle Y$, and $\displaystyle Z$ are the midpoints of their respective sides, $\displaystyle AZ = BZ= XD = BY = 1$, as shown in this diagram.

Pentagon 3

The area of $\bigtriangleup BZY$ $\displaystyle \bigtriangleup BZY$, it being a right triangle, is half the product of the lengths of its legs:

$\frac{1}{2} \cdot BZ \cdot BY = \frac{1}{2} \cdot 1 \cdot 1 = \frac{1}{2}$ $\displaystyle \frac{1}{2} \cdot BZ \cdot BY = \frac{1}{2} \cdot 1 \cdot 1 = \frac{1}{2}$

The area of $\bigtriangleup ZXD$ $\displaystyle \bigtriangleup ZXD$ is half the product of the length of a base and the height. Using $\overline{XD}$ $\displaystyle \overline{XD}$ as the base, and $\overline{AZ}$ $\displaystyle \overline{AZ}$ as an altitude:

$\frac{1}{2} \cdot XD \cdot AZ= \frac{1}{2} \cdot 1 \cdot 1 = \frac{1}{2}$ $\displaystyle \frac{1}{2} \cdot XD \cdot AZ= \frac{1}{2} \cdot 1 \cdot 1 = \frac{1}{2}$

The two triangles have the same area.

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

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

All ISEE Upper Level Quantitative Resources

Example Questions

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

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

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

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

All ISEE Upper Level Quantitative Resources

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