ISEE Upper Level Quantitative Reasoning - Plane Geometry

A rectangle is two feet longer than it is wide; its perimeter is 11 feet. What is its area in square inches?

$945 \textrm{ in}^{2}$

$513 \textrm{ in}^{2}$

$4,212 \textrm{ in}^{2}$

$3,780 \textrm{ in}^{2}$

It is impossible to determine the area from the information given

Explanation

The length of the rectangle is 2 feet, or 24 inches, greater than the width, so, if is the width in inches, is the length in inches.

The perimeter of the rectangle is 11 feet, or $11 \times 12= 132$ inches. The perimeter, in terms of length and width, is , so we can set up the equation:

$4 W \div 4 =84 \div 4$

The width is 21 inches, and the length is 45 inches. The area is their product:

$21\times 45 = 945$ square inches.

Find the perimeter of a pentagon with a side of length 12in.

$60\text{in}$

$72\text{in}$

$96\text{in}$

$48\text{in}$

$58\text{in}$

Explanation

To find the perimeter of a pentagon, we will use the following formula:

$P = 5 \cdot a$

where a is the length of one side of the pentagon.

Now, we know the length of one side of the pentagon is 12in.

Knowing this, we can substitute into the formula. We get

$P = 5 \cdot 12\text{in}$

$P = 60\text{in}$

A regular octagon has perimeter one meter. Which is the greater quantity?

(A) The length of one side

(B) 125 millimeters

(A) and (B) are equal

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

(A) is greater

(B) is greater

Explanation

A regular octagon has eight sides of equal length. The perimeter of this octagon is one meter, which is equal to 1,000 millimeters; each side, therefore, has length

$1,000 \div 8 = 125$ millimeters

making the quantities equal.

A rectangle has perimeter 140 inches and area 1,200 square inches. Which is the greater quantity?

(A) The length of a diagonal of the rectangle.

(B) 4 feet

(A) is greater

(B) is greater

(A) and (B) are equal

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

Explanation

Let and be the dimensions of the rectangle. Then

and, subsequently,

Since the product of the length and width is the area, we are looking for two numbers whose sum is 70 and whose product is 1,200; through trial and error, they are found to be 30 and 40. We can assign either to be and the other to be since the result is the same.

The length of a diagonal of the rectangle can be found by applying the Pythagorean Theorem:

$D = \sqrt{L^{2}+ W^{2}}$

$D = \sqrt{30^{2}+ 40^{2}} = \sqrt{900+1,600} = \sqrt{2,500} = 50$

A diagonal is 50 inches long; since 4 feet are equivalent to 48 inches, (A) is the greater quantity.

Find the perimeter of a pentagon with a side of length 12in.

$60\text{in}$

$72\text{in}$

$96\text{in}$

$48\text{in}$

$58\text{in}$

Explanation

To find the perimeter of a pentagon, we will use the following formula:

$P = 5 \cdot a$

where a is the length of one side of the pentagon.

Now, we know the length of one side of the pentagon is 12in.

Knowing this, we can substitute into the formula. We get

$P = 5 \cdot 12\text{in}$

$P = 60\text{in}$

Trapezoid

In the above figure, $\overline{XY}$ is the midsegment of Trapezoid .

Which is the greater quantity?

(a) Three times the area of Trapezoid

(b) Twice the area of Trapezoid

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

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

Explanation

Midsegment $\overline{XY}$ divides Trapezoid into two trapezoids of the same height, which we will call ; the length of the midsegment is half sum of the lengths of the bases:

$XY = \frac{1}{2} (12 + 36) = \frac{1}{2} \cdot 48= 24$

The area of a trapezoid is one half multiplied by its height multiplied by the sum of the lengths of its bases. Therefore, the area of Trapezoid is

$\frac{1}{2} \cdot h \cdot (TR+XY ) = \frac{1}{2} \cdot h \cdot (12+24) = \frac{1}{2} \cdot 36 \cdot h = 18 h$ .

Three times this is

$3 \cdot 18h = 54 h$ .

The area of Trapezoid is, similarly,

$\frac{1}{2} \cdot h \cdot (XY +AP ) = \frac{1}{2} \cdot h \cdot (24+36 ) = \frac{1}{2} \cdot 60 \cdot h = 30 h$

Twice this is

$2 \cdot 30 h = 60 h$ .

That makes (b) the greater quantity.

Trapezoid

In the above figure, $\overline{XY}$ is the midsegment of Trapezoid . What percent of Trapezoid has been shaded in?

$37 \frac{1}{2} %$

$33 \frac{1}{3} %$

Explanation

Midsegment $\overline{XY}$ divides Trapezoid into two trapezoids of the same height, which we will call ; the length of the midsegment is half sum of the lengths of the bases:

$XY = \frac{1}{2} (12 + 36) = \frac{1}{2} \cdot 48= 24$

The area of a trapezoid is one half multiplied by its height multiplied by the sum of the lengths of its bases. Therefore, the area of Trapezoid - the shaded trapezoid - is

$\frac{1}{2} \cdot h \cdot (TR+XY ) = \frac{1}{2} \cdot h \cdot (12+24) = \frac{1}{2} \cdot 36 \cdot h = 18 h$

The area of Trapezoid is

$\frac{1}{2} \cdot 2 h \cdot (TR+AP) = \frac{1}{2} \cdot 2 \cdot h \cdot (12+36) = \frac{1}{2} \cdot 2 \cdot 48 \cdot h = 48 h$

The percent of Trapezoid that is shaded in is

$\frac{18h}{48h} \cdot 100 % = \frac{18}{48} \cdot 100 % = 37 \frac{1}{2} %$

Sector TYP occupies 43% of a circle. Find the degree measure of angle TYP.

$154.8^{\circ}$

$15.5^{\circ}$

$430^{\circ}$

$366^{\circ}$

Explanation

Sector TYP occupies 43% of a circle. Find the degree measure of angle TYP.

Use the following formula and solve for x:

$\frac{x}{360^{\circ}}100=43%$

Begin by dividing over the 100

$\frac{x}{360^{\circ}}=.43$

Then multiply by 360

$x=.43*360^{\circ}=154.8^{\circ}$

One side of a regular pentagon is 20% longer than one side of a regular hexagon. Which is the greater quantity?

(A) The perimeter of the pentagon

(B) The perimeter of the hexagon

(A) and (B) are equal

(B) is greater

(A) is greater

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

Explanation

Let be the length of one side of the hexagon. Then its perimeter is .

Each side of the pentagon is 20% greater than this length, or

The perimeter is five times this, or $5 \cdot 1.2s = 6s$ .

The perimeters are the same.

The perimeter of a square is one yard. Which is the greater quantity?

(a) The area of the square

(b) $\frac{1}{2}$ square foot

(a) is greater.

(b) is greater.

(a) and (b) are equal.

It is impossible to tell form the information given.

Explanation

One yard is equal to three feet, so the length of one side of a square with this perimeter is $\frac{3}{4}$ feet. The area of the square is $\frac{3}{4}\times \frac{3}{4} = \frac{9}{16}$ square feet. $\frac{9}{16} > \frac{1}{2}$ , making (a) greater.

Plane Geometry

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