Pentagons - ISEE Upper Level Quantitative Reasoning

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Question

The above diagram depicts trapezoid . Which is the greater quantity?

(a) $m \angle T + m \angle P$

(b) $m \angle R + m \angle A$

Answer

$\overline{TR} \parallel \overline{PA}$ ; $\angle T$ and $\angle P$ are same-side interior angles, as are $\angle R$ and $\angle A$ .

The Same-Side Interior Angles Theorem states that if two parallel lines are crossed by a transversal, then the sum of the measures of a pair of same-side interior angles is always $180^{\circ}$ .

Therefore, $m \angle T + m \angle P = 180= m \angle R + m \angle A$ , making the two quantities equal.

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Question

Given Trapezoid , where $\overline{TR} \parallel \overline{PA}$ . Also, $m \angle A > m\angle P$

Which is the greater quantity?

(a) $m\angle T$

(b) $m \angle R$

Answer

$\angle T$ and $\angle P$ are same-side interior angles, as are $\angle R$ and $\angle A$ .

The Same-Side Interior Angles Theorem states that if two parallel lines are crossed by a transversal, then the sum of the measures of a pair of same-side interior angles is always $180^{\circ}$ . Therefore,

$m \angle T + m \angle P = 180$ , or $m \angle P = 180 - m \angle T$

$m \angle R + m \angle A = 180$ , or $m \angle A = 180 - m \angle R$

Substitute:

$m \angle A > m\angle P$

$180 - m \angle R > 180 - m\angle T$

$180 - m \angle R -180 > 180 - m\angle T-180$

$- m \angle R > - m\angle T$

$- (- m \angle R )<-\left ( - m\angle T \right )$

$m \angle R < m\angle T$

(a) is the greater quantity

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Question

Consider trapezoid , where $\overline{TR} \parallel \overline{PA}$ . Also, $\angle A$ is acute and $\angle P$ is obtuse.

Which is the greater quantity?

(a) $m \angle T$

(b) $m \angle R$

Answer

$\angle T$ and $\angle P$ are same-side interior angles, as are $\angle R$ and $\angle A$ .

The Same-Side Interior Angles Theorem states that if two parallel lines are crossed by a transversal, then same-side interior angles are supplementary. A pair of supplementary angles comprises either two right angles, or one acute angle and one obtuse angle. Since $\angle A$ is acute and $\angle P$ is obtuse, $\angle R$ is obtuse and $\angle T$ is acute. Therefore $\angle R$ the greater measure of the two, making (b) greater.

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Question

Which quantity is greater?

(a) The perimeter of the above trapezoid

(b) The perimeter of a rectangle with length and width and , respectively.

Answer

The perimeter of a rectangle is twice the sum of its length and its width:

Since the height of the trapezoid in the figure is , both of its legs must have length greater than or equal to . But for a leg to be of length, it must be perpendicular to the bases. Since perpendicularity of both legs would make the trapezoid a rectangle - which it cannot be - it follows that both legs cannot be of length . Therefore, the perimeter of the trapezoid is:

The perimeter of the trapezoid must be greater than that of the rectangle.

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Question

Figure NOT drawn to scale.

In the above figure, $\overline{XY}$ is the midsegment of isosceles Trapezoid . Also, $TX = \frac{1}{2} XY$ .

What is the perimeter of Trapezoid ?

Answer

The length of the midsegment of a trapezoid is half sum of the lengths of the bases, so

$XY = \frac{1}{2} (12 + 40) = \frac{1}{2} \cdot 52 = 26$ .

Also, by definition, since Trapezoid is isosceles, . The midsegment divides both legs of Trapezoid into congruent segments; combining these facts:

$TX = XP = \frac{1}{2} TP = \frac{1}{2} RA = RY = YA$

$TX = \frac{1}{2} XY = \frac{1}{2} \cdot 26 = 13$ .

, so the perimeter of Trapezoid is

.

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Question

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

Which is the greater quantity?

(a) Twice the perimeter of Trapezoid

(b) The perimeter of Trapezoid

Answer

The midsegment of a trapezoid bisects both of its legs, so

and .

For reasons that will be apparent later, we will set

Also, 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 perimeter of Trapezoid is

Twice this is

The perimeter of Trapezoid is

and , so , making (a) the greater quantity.

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Question

Refer to the above figure, which shows a parallelogram. What is equal to?

Answer

The sum of two consecutive angles of a parallelogram is $180^{\circ }$ .

157 is the correct choice.

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Question

Which of the following can be the measures of the four angles of a parallelogram?

Answer

Opposite angles of a parallelogram must have the same measure, so the correct choice must have two pairs, each of the same angle measure. We can therefore eliminate $38^{\circ }, 142 ^{\circ }, 28^{\circ }, 152 ^{\circ }$ and $102^{\circ }, 94 ^{\circ }, 86^{\circ }, 78 ^{\circ }$ as choices.

Also, the sum of the measures of the angles of any quadrilateral must be $360 ^{\circ }$ , so we add the angle measures of the remaining choices:

$88^{\circ }, 97 ^{\circ }, 88^{\circ }, 97 ^{\circ }$ :

$88^{\circ } + 97 ^{\circ } + 88^{\circ }+ 97 ^{\circ } = 370$ , so we can eliminate this choice.

$77^{\circ }, 133 ^{\circ }, 77^{\circ }, 133 ^{\circ }$ :

$77^{\circ } + 133 ^{\circ } + 77^{\circ }+ 133 ^{\circ } =420$ , so we can eliminate this choice.

$57^{\circ }, 123 ^{\circ }, 57^{\circ }, 123 ^{\circ }$

$57^{\circ } +123 ^{\circ }+57^{\circ }+123 ^{\circ } = 360$ ; this is the correct choice.

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Question

In Parallelogram , $m \angle A = (3x)^{\circ }$ and $m \angle C = (2y)^{\circ }$ .

Which is the greater quantity?

(a)

(b)

Answer

In Parallelogram , $\angle A$ and $\angle C$ are opposite angles and are therefore congruent. This means that

$m \angle A = m \angle C$

$3x \div 3 = 2y \div 3$

$x = \frac{2}{3}y$

Both are positive, so .

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Question

In the above parallelogram, $\angle 1$ is acute. Which is the greater quantity?

(A) The perimeter of the parallelogram

(B) 46 inches

Answer

The measure of $\angle 1$ is actually irrelevant. The perimeter of the parallelogram is the sum of its four sides; since opposite sides of a parallelogram have the same length, the perimeter is

inches,

making the quantities equal.

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Question

Parallelogram A is below:

Parallelogram B is below:

Note: These figures are NOT drawn to scale.

Refer to the parallelograms above. Which is the greater quantity?

(A) The perimeter of parallelogram A

(B) The perimeter of parallelogram B

Answer

The perimeter of a parallelogram is the sum of its sidelengths; its height is irrelevant. Also, opposite sides of a parallelogram are congruent.

The perimeter of parallelogram A is

inches;

The perimeter of parallelogram B is

inches.

(A) is greater.

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Question

Figure NOT drawn to scale.

The above figure depicts Rhombus with and .

Give the perimeter of Rhombus .

Answer

All four sides of a rhombus have the same length, so we can find the perimeter of Rhombus by taking the length of one side and multiplying it by four. Since , the perimeter is four times this, or $4 \cdot AB = 4 \cdot 32 = 128$ .

Note that the length of $\overline{AY}$ is actually irrelevant to the problem.

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Question

Rhombus has two diagonals that intersect at point ; $\angle RHO = 75 ^{\circ }$ . Which is the greater quantity?

(a) $m \angle RPH$

(b) $m \angle HPO$

Answer

The diagonals of a rhombus always intersect at right angles, so $m \angle RPH = m \angle HPO = 90 ^{\circ }$ . The measures of the interior angles of the rhombus are irrelevant.

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Question

A rhombus has diagonals of length two and one-half feet and six feet. Which is the greater quantity?

(A) The perimeter of the rhombus

(B) Four yards

Answer

It will be easier to look at these measurements as inches for the time being:

$2 \frac{1}{2} \times 12 = 30$ and $6 \times 12 = 72$ , so these are the lengths of the diagonals in inches.

The diagonals of a rhombus are each other's perpendicular bisector, so, as can be seen in the diagram below, one side of a rhombus and one half of each diagonal form a right triangle. If we let be the length of one side of the rhombus, then this is the hypotenuse of that right triangle; its legs are one-half the lengths of the diagonals, or 15 and 36 inches.

By the Pythagorean Theorem,

$s = \sqrt{ 15 ^{2} + 36^{2}} = \sqrt{ 225 + 1,296} = \sqrt{ 1,521} = 39$

Each side of the rhombus measures 39 inches, and its perimeter is

$4 \times 39 = 156$ inches.

Four yards is equal to $4 \times 36 = 144$ inches, so (A) is greater.

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Question

A rectangle has a width of 2_x_. If the length is five more than 150% of the width, what is the perimeter of the rectangle?

Answer

Given that w = 2_x_ and l = 1.5_w_ + 5, a substitution will show that l = 1.5(2_x_) + 5 = 3_x_ + 5.

P = 2_w_ + 2_l_ = 2(2_x_) + 2(3_x_ + 5) = 4_x_ + 6_x_ + 10 = 10_x_ + 10 = 10(x + 1)

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Question

A rectangle has length 72 inches and width 36 inches. What is its perimeter?

Answer

The perimeter of a rectangle is equal to twice the sum of its length and its width, which here would be, in inches,

$2 (72 + 36) = 2 \times 108 = 216 \textrm{ in}$ .

$216 \textrm{ in} = 216\div 12 = 18 \textrm{ ft} = 18 \div 3 = 6 \textrm{ yd}$

$216 \textrm{ in} = 18 \textrm{ ft} = \frac{18}{5,280} = \frac{18\div 6}{5,280\div 6}= \frac{3}{880}\textrm{ mi}$

Therefore, the correct choice is that all four measurements are equal to the perimeter.

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Question

Which quantity is greater?

(a) The perimeter of a square with area 10,000 square centimeters

(b) The perimeter of a rectangle with area 8,000 square centimeters

Answer

A square with area 10,000 square centimeters has sidelength $\sqrt{10,000} = 100$ centimeters, and perimeter $4 \times 100 = 400$ centimeters.

Not enough information is given about the rectangle with area 8,000 square centimeters to determine its perimeter. For example, if its dimensions are 100 centimeters by 80 centimeters, its perimeter is centimeters. If the dimensions are 200 centimeters by 40 centimeters, its perimeter is centimeters. Both cases are consistent with the conditions of the problem, yet one makes (a) greater and one makes (b) greater.

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Question

Which is the greater quantity?

(a) The perimeter of the rectangle on the coordinate plane with vertices

(b) The perimeter of the rectangle on the coordinate plane with vertices

Answer

(a) The first rectangle has width and height ; its perimeter is

.

(b) The second rectangle has width and height ; its perimeter is

.

For the first rectangle to have a greater perimeter, it is necessary for

, or equivalently,

.

$4y\div 4> 2x\div 4$

$y > \frac{1}{2}x$

We do not know the relative values of and , however, so we cannot compare their perimeters.

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Question

The sum of the lengths of three sides of a square is one meter. Give the perimeter of the square in millimeters.

Answer

A square has four sides of the same length.

The sum of the lengths of three sides of a square is one meter, which is equal to 1,000 millimeters, so each side has length

$1,000 \div 3 = 333 \frac{1}{3}$ millimeters,

and the perimeter is four times this, or

$4 \times 333 \frac{1}{3} = 1,333 \frac{1}{3}$ millimeters.

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Question

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

Answer

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.

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