ISEE Lower Level Quantitative : Geometry

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

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Example Questions

Example Question #1 : How To Find A Square On A Coordinate Plane

Square \(\displaystyle ABCD\) has coordinate points: \(\displaystyle A(0,0)\)\(\displaystyle B(5,0)\),\(\displaystyle C(5,5)\)\(\displaystyle D(0,5)\). Find the perimeter of square \(\displaystyle ABCD\).

Possible Answers:

\(\displaystyle 18\)

\(\displaystyle 15\)

\(\displaystyle 25\)

\(\displaystyle 20\)

Correct answer:

\(\displaystyle 20\)

Explanation:

The perimeter of square \(\displaystyle ABCD\) can be found by applying the formula: 

\(\displaystyle 4S\), where \(\displaystyle S=\) the length of one side of the square.

To find the length of the square look at the coordinates:

\(\displaystyle A=(0,0), B=(5,0)\rightarrow S=5\)

Thus, 
\(\displaystyle S=5\)
\(\displaystyle p=4\times5=20\)

Example Question #1 : How To Find A Square On A Coordinate Plane

The points with coordinates \(\displaystyle (7, 5), (5, 3), (5, 5), (7, 3)\) are the vertices of which kind of quadrilateral?

Possible Answers:

Kite

Square

Parallelogram

Trapezoid

Correct answer:

Square

Explanation:

1

If the points are plotted on to a graph, you should notice that the points form a square with side lengths of \(\displaystyle 2\).

Example Question #1 : Trapezoids

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:

\(\displaystyle 8.5\)

\(\displaystyle 8\)

\(\displaystyle 7\)

\(\displaystyle 3\)

\(\displaystyle 6\)

Correct answer:

\(\displaystyle 3\)

Explanation:

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

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

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

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

\(\displaystyle 3\)

Example Question #1 : Quadrilaterals

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

The find the area of a trapezoid, use:

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

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

Yard

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

Possible Answers:

\(\displaystyle 6,125 \textrm{ m}^{2}\)

\(\displaystyle 10,325 \textrm{ m}^{2}\)

\(\displaystyle 8,925 \textrm{ m}^{2}\)

\(\displaystyle 7,525 \textrm{ m}^{2}\)

\(\displaystyle 11,350 \textrm{ m}^{2}\)

Correct answer:

\(\displaystyle 7,525 \textrm{ m}^{2}\)

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

\(\displaystyle 105 \times 85 = 8,925\) 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

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

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

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

Example Question #1 : Trapezoids

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:

\(\displaystyle 41,000 \textrm{ ft}^{2}\)

\(\displaystyle 46,500 \textrm{ ft}^{2}\)

\(\displaystyle 35,600 \textrm{ ft}^{2}\)

\(\displaystyle 16,000 \textrm{ ft}^{2}\)

\(\displaystyle 20,500\textrm{ ft}^{2}\)

Correct answer:

\(\displaystyle 20,500\textrm{ ft}^{2}\)

Explanation:

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

\(\displaystyle \frac{1}{2 } \times 200 \times 250 = 25,000\) 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

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

 

Therefore, Mr. Smith retains a parcel of area

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

Example Question #233 : Geometry

Screenshot_2015-03-29_at_3.04.28_pm

What is the area of the above trapezoid?

Possible Answers:

\(\displaystyle 35\:in^2\)

\(\displaystyle 40\:in^2\)

\(\displaystyle 45\:in^2\)

\(\displaystyle 63\:in^2\)

\(\displaystyle 52\:in^2\)

Correct answer:

\(\displaystyle 40\:in^2\)

Explanation:

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

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:

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

Example Question #1 : Quadrilaterals

Screenshot_2015-03-29_at_3.18.30_pm

What is the area of the above trapezoid?

Possible Answers:

\(\displaystyle 40\:cm^2\)

\(\displaystyle 50\:cm^2\)

\(\displaystyle 96\:cm^2\)

\(\displaystyle 35\:cm^2\)

\(\displaystyle 60\:cm^2\)

Correct answer:

\(\displaystyle 50\:cm^2\)

Explanation:

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

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

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

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

Example Question #1 : Quadrilaterals

Find the perimeter of the following trapezoid. 

8

Possible Answers:

\(\displaystyle 12\)

\(\displaystyle 23\)

\(\displaystyle 13\)

\(\displaystyle 21\)

Correct answer:

\(\displaystyle 21\)

Explanation:

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

\(\displaystyle 6+3+5+7=21\)

Example Question #2 : Quadrilaterals

Find the perimeter of the following trapezoid. 

9

Possible Answers:

\(\displaystyle 33\)

\(\displaystyle 19\)

\(\displaystyle 29\)

\(\displaystyle 18\)

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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