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ISEE Lower Level Quantitative : Plane Geometry

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

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All ISEE Lower Level Quantitative Resources

20 Diagnostic Tests 114 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Quadrilaterals

Trapezoid3

What is the area of a trapezoid if its height is 1, its long base is 4, and its short base is 2?

Possible Answers:

$\displaystyle 6$

$\displaystyle 7$

8.5 $\displaystyle 8.5$

$\displaystyle 8$

$\displaystyle 3$

Correct answer:

$\displaystyle 3$

Explanation:

The area of a trapezoid is given by the formula $\frac{1}{2} *height* (base_{long} + base_{short})$ $\displaystyle \frac{1}{2} *height* (base_{long} + base_{short})$.

We know that the height is 1, the long base is 4, and the short base is 2.

$\frac{1}{2} *height* (base_{long} + base_{short})=\frac{1}{2} *(1)* ((4) + (2))$ $\displaystyle \frac{1}{2} *height* (base_{long} + base_{short})=\frac{1}{2} *(1)* ((4) + (2))$

$\frac{1}{2}*(6)$ $\displaystyle \frac{1}{2}*(6)$

$\displaystyle 3$

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Example Question #1 : Plane Geometry

Which formula would you use to find the area of a trapezoid?

Possible Answers:

$area=\left ( \frac{h}{2} \right )\times b1+b2$ $\displaystyle area=\left ( \frac{h}{2} \right )\times b1+b2$

$area=\left ( \frac{b1+b2}{2} \right )\times h$ $\displaystyle area=\left ( \frac{b1+b2}{2} \right )\times h$

$area=\left ( \frac{b1+b2}{h} \right )\times 2$ $\displaystyle area=\left ( \frac{b1+b2}{h} \right )\times 2$

$area=\left ( \frac{b1\times b2}{2} \right )\times h$ $\displaystyle area=\left ( \frac{b1\times b2}{2} \right )\times h$

Correct answer:

$area=\left ( \frac{b1+b2}{2} \right )\times h$ $\displaystyle area=\left ( \frac{b1+b2}{2} \right )\times h$

Explanation:

The find the area of a trapezoid, use:

$area=\left ( \frac{b1+b2}{2} \right )\times h$ $\displaystyle area=\left ( \frac{b1+b2}{2} \right )\times h$

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Example Question #1 : Plane Geometry

Yard

The above diagram shows a trapezoidal home within a rectangular yard. What is the area of the yard?

Possible Answers:

$7,525 \textrm{ m}^{2}$ $\displaystyle 7,525 \textrm{ m}^{2}$

$6,125 \textrm{ m}^{2}$ $\displaystyle 6,125 \textrm{ m}^{2}$

$8,925 \textrm{ m}^{2}$ $\displaystyle 8,925 \textrm{ m}^{2}$

$10,325 \textrm{ m}^{2}$ $\displaystyle 10,325 \textrm{ m}^{2}$

$11,350 \textrm{ m}^{2}$ $\displaystyle 11,350 \textrm{ m}^{2}$

Correct answer:

$7,525 \textrm{ m}^{2}$ $\displaystyle 7,525 \textrm{ m}^{2}$

Explanation:

The area of the yard is the area of the small trapezoid subtracted from that of the large rectangle. The area of a rectangle is the product of its length and its height, so the large rectangle has area

$105 \times 85 = 8,925$ $\displaystyle 105 \times 85 = 8,925$ square meters.

The area of a trapezoid is half the product of its height and the sum of its two parallel sides (bases), so the small trapezoid has area

$\frac{1}{2} \times 40 \times (30 + 40) = \frac{1}{2} \times 40 \times 70 = 1,400$ $\displaystyle \frac{1}{2} \times 40 \times (30 + 40) = \frac{1}{2} \times 40 \times 70 = 1,400$ square meters.

The area of the yard is the difference of the two:

$\displaystyle 8,925- 1,400 = 7,525$ square meters.

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

Yard

Note: Figure NOT drawn to scale.

Mr. Smith owns the triangular piece of land seen in the above diagram. He sells the trapezoidal parcel shown at bottom right to his brother. What is the area of the land he retains?

Possible Answers:

$35,600 \textrm{ ft}^{2}$ $\displaystyle 35,600 \textrm{ ft}^{2}$

$41,000 \textrm{ ft}^{2}$ $\displaystyle 41,000 \textrm{ ft}^{2}$

$20,500\textrm{ ft}^{2}$ $\displaystyle 20,500\textrm{ ft}^{2}$

$16,000 \textrm{ ft}^{2}$ $\displaystyle 16,000 \textrm{ ft}^{2}$

$46,500 \textrm{ ft}^{2}$ $\displaystyle 46,500 \textrm{ ft}^{2}$

Correct answer:

$20,500\textrm{ ft}^{2}$ $\displaystyle 20,500\textrm{ ft}^{2}$

Explanation:

The area of a triangle is half the product of its base and its height, so Mr. Smith's parcel originally had area

$\frac{1}{2 } \times 200 \times 250 = 25,000$ $\displaystyle \frac{1}{2 } \times 200 \times 250 = 25,000$ square feet.

The area of a trapezoid is the half product of its height and the sum of its two parallel sides (bases), so the portion Mr. Smith sold to his brother has area

$\frac{1}{2} \times 50 \times (80 + 100) = 4,500$ $\displaystyle \frac{1}{2} \times 50 \times (80 + 100) = 4,500$ square feet.

Therefore, Mr. Smith retains a parcel of area

$\displaystyle 25,000 - 4,500 = 20,500$ square feet.

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Example Question #1 : Plane Geometry

Screenshot_2015-03-29_at_3.04.28_pm

What is the area of the above trapezoid?

Possible Answers:

$63\:in^2$ $\displaystyle 63\:in^2$

$40\:in^2$ $\displaystyle 40\:in^2$

$35\:in^2$ $\displaystyle 35\:in^2$

$52\:in^2$ $\displaystyle 52\:in^2$

$45\:in^2$ $\displaystyle 45\:in^2$

Correct answer:

$40\:in^2$ $\displaystyle 40\:in^2$

Explanation:

The formula for the area of a trapezoid is $\frac{base_1 + base_2}{2}\cdot height$ $\displaystyle \frac{base_1 + base_2}{2}\cdot height$

In other words, find the average of the bases and multiply by the height. Substituting in the values of the bases for the given trapezoid, $\displaystyle 9$ and $\displaystyle 7$, you get:

$\frac{9\:in+7\:in}{2}\cdot 5\:in = 8\:in\cdot 5\:in=40\:in^2$ $\displaystyle \frac{9\:in+7\:in}{2}\cdot 5\:in = 8\:in\cdot 5\:in=40\:in^2$

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Example Question #6 : Plane Geometry

Screenshot_2015-03-29_at_3.18.30_pm

What is the area of the above trapezoid?

Possible Answers:

$40\:cm^2$ $\displaystyle 40\:cm^2$

$50\:cm^2$ $\displaystyle 50\:cm^2$

$35\:cm^2$ $\displaystyle 35\:cm^2$

$96\:cm^2$ $\displaystyle 96\:cm^2$

$60\:cm^2$ $\displaystyle 60\:cm^2$

Correct answer:

$50\:cm^2$ $\displaystyle 50\:cm^2$

Explanation:

The formula for the area of a trapezoid is $\frac{base_1 + base_2}{2}\cdot height$ $\displaystyle \frac{base_1 + base_2}{2}\cdot height$

In other words, find the average of the bases and multiply by the height. Substituting the values of the bases of the given trapezoid, $8\:cm$ $\displaystyle 8\:cm$ and $12\:cm$ $\displaystyle 12\:cm$, into the equation, you get:

$\frac{8\:cm+12\:cm}{2}\cdot 5\:cm = 10\:cm\cdot 5\:cm=50\:cm^2$ $\displaystyle \frac{8\:cm+12\:cm}{2}\cdot 5\:cm = 10\:cm\cdot 5\:cm=50\:cm^2$

The area of the trapezoid is thus $50\:cm^2$ $\displaystyle 50\:cm^2$.

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Example Question #1 : Trapezoids

Find the perimeter of the following trapezoid.

Possible Answers:

$\displaystyle 23$

$\displaystyle 13$

$\displaystyle 12$

$\displaystyle 21$

Correct answer:

$\displaystyle 21$

Explanation:

In order to find the perimeter of the trapezoid, we have to add up all the outer sides:

$\displaystyle 6+3+5+7=21$

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Example Question #2 : Trapezoids

Find the perimeter of the following trapezoid.

Possible Answers:

$\displaystyle 19$

$\displaystyle 29$

$\displaystyle 18$

$\displaystyle 33$

Correct answer:

$\displaystyle 29$

Explanation:

In order to find the perimeter we must add up all the outer sides of the trapezoid.

$\displaystyle 5+9+7+8=29$

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Example Question #241 : Geometry

Daria found the perimeter of the below trapezoid to be 29 inches.

Screenshot_2015-03-29_at_3.03.48_pm

What is the length of the missing side?

Possible Answers:

$8.5\:in$ $\displaystyle 8.5\:in$

$7\:in$ $\displaystyle 7\:in$

$6\:in$ $\displaystyle 6\:in$

$9\:in$ $\displaystyle 9\:in$

$5\:in$ $\displaystyle 5\:in$

Correct answer:

$6\:in$ $\displaystyle 6\:in$

Explanation:

To find the perimeter of a trapezoid, add up the lengths of its four sides. In the given trapezoid,

$x+7\:in+7\:in+9\:in=29\:in$ $\displaystyle x+7\:in+7\:in+9\:in=29\:in$

To find the missing side, combine like terms and solve for $\displaystyle x$:

x+23=29 $\displaystyle x+23=29$

$\displaystyle x+23-23=29-23$

$\displaystyle x=29-23=6$

The missing side of the trapezoid is $6\:in$ $\displaystyle 6\:in$ long.

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Example Question #1 : Plane Geometry

Screenshot_2015-03-29_at_3.04.28_pm

What is the perimeter of the above trapezoid?

Possible Answers:

$30\:in$ $\displaystyle 30\:in$

$29\:in$ $\displaystyle 29\:in$

$24\:in$ $\displaystyle 24\:in$

$28\:in$ $\displaystyle 28\:in$

$23\:in$ $\displaystyle 23\:in$

Correct answer:

$28\:in$ $\displaystyle 28\:in$

Explanation:

To find the perimeter of a trapezoid, add up the lengths of the four sides. In the given trapezoid,

$9\:in+7\:in+7\:in+5\:in=28\:in$ $\displaystyle 9\:in+7\:in+7\:in+5\:in=28\:in$

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All ISEE Lower Level Quantitative Resources

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ISEE Lower Level Quantitative : Plane Geometry

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

All ISEE Lower Level Quantitative Resources

Example Questions

Example Question #1 : Quadrilaterals

Example Question #1 : Plane Geometry

Example Question #1 : Plane Geometry

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

Example Question #1 : Plane Geometry

Example Question #6 : Plane Geometry

Example Question #1 : Trapezoids

Example Question #2 : Trapezoids

Example Question #241 : Geometry

Example Question #1 : Plane Geometry

All ISEE Lower Level Quantitative Resources

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