Quadrilaterals - SAT Math

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Question

Find the perimeter of a square whose side length is 5.

Answer

To solve, simply use the formula for the perimeter of a square. Thus,

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Question

Find the area of a parallelogram with length 6 and height 5.

Answer

To solve, simply use the formula for the area of a parallelogram. Thus,

Don't let a parallelogram fool you. It is just like a rectangle but on a slant. As long as you are given the height, and not the slant side length, you can use the same formula.

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Question

George wants to paint the walls in his room blue. The ceilings are 10 ft tall and a carpet 12 ft by 15 ft covers the floor. One gallon of paint covers 400 $ft^{2}$ and costs $40. One quart of paint covers 100 $ft^{2}$ and costs $15. How much money will he spend on the blue paint?

Answer

The area of the walls is given by $A = 2wh + 2lh = 2(12)(10) + 2(15)(10) = 540 ft^{2}$

One gallon of paint covers 400 $ft^{2}$ and the remaining 140 $ft^{2}$ would be covered by two quarts.

So one gallon and two quarts of paint would cost

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Question

A trapezoid has a base of length 4, another base of length s, and a height of length s. A square has sides of length s. What is the value of s such that the area of the trapezoid and the area of the square are equal?

Answer

In general, the formula for the area of a trapezoid is (1/2)(a + b)(h), where a and b are the lengths of the bases, and h is the length of the height. Thus, we can write the area for the trapezoid given in the problem as follows:

area of trapezoid = (1/2)(4 + s)(s)

Similarly, the area of a square with sides of length a is given by _a_2. Thus, the area of the square given in the problem is _s_2.

We now can set the area of the trapezoid equal to the area of the square and solve for s.

(1/2)(4 + s)(s) = _s_2

Multiply both sides by 2 to eliminate the 1/2.

(4 + s)(s) = 2_s_2

Distribute the s on the left.

4_s_ + _s_2 = 2_s_2

Subtract _s_2 from both sides.

4_s_ = _s_2

Because s must be a positive number, we can divide both sides by s.

4 = s

This means the value of s must be 4.

The answer is 4.

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Question

Find the area of a trapezoid given bases of length 1 and 2 and height of 2.

Answer

To solve, simply use the formula for the area of a trapezoid. Thus,

$A=\frac{1}{2}(b1+b2)h=\frac{1}{2}(1+2)(2)=3$

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Question

The above figure shows Square . is the midpoint of $\overline{AB}$ ; is the midpoint of $\overline{CD}$ ; is the midpoint of $\overline{CN}$ . Construct $\overline{MO}$ .

If Square has area , what is the area of Quadrilateral ?

Answer

Let be the common sidelength of the square. The area of the square is $X = t^{2}$ .

Construct segment $\overline{MN}$ . This divides the square into Rectangle and a right triangle. The dimensions of Rectangle are

and

$ND = \frac{1}{2} CD = \frac{1}{2} t$ .

The area of Rectangle s the product of these dimensions:

$A_{1} = AD \cdot ND = t \cdot \frac{1}{2} t = \frac{1}{2} t ^{2}$

The lengths of the legs of Right $\bigtriangleup MNO$ are

and

$NO = \frac{1}{2} CN = \frac{1}{2} \cdot \frac{1}{2} CD = \frac{1}{2} \cdot \frac{1}{2} t = \frac{1}{4} t$

The area of this right triangle is half the product of these lengths, or

$A_{2} = \frac{1}{2} \cdot MN \cdot NO = \frac{1}{2} \cdot t \cdot \frac{1}{4} t = \frac{1}{8} t ^{2}$

This is seen below:

The sum of these areas is the area of Quadrilateral

$A = A_{1}+ A_{2}= t^{2}+\frac{1}{8}t ^{2} = \frac{5}{8} t ^{2}$ .

Substituting for $t^{2}$ , the area is $\frac{5}{8} X$ .

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Question

The above figure shows Square . is the midpoint of $\overline{AB}$ ; is the midpoint of $\overline{CD}$ ; is the midpoint of $\overline{CN}$ . Construct $\overline{MO}$ .

. Which of the following expresses the length of $\overline{MO}$ in terms of ?

Answer

Construct $\overline{MN}$ as shown in the diagram below:

Quadrilateral is a rectangle, so opposite sides are congruent. Therefore, .

Since is the midpoint of $\overline{CD}$ ,

$CN = \frac{1}{2} CD = \frac{1}{2} t$

Since is the midpoint of $\overline{CN}$ ,

$ON = \frac{1}{2} CN = \frac{1}{2} \cdot \frac{1}{2} t = \frac{1}{4} t$ .

$\bigtriangleup MNO$ is a right triangle, so, by the Pythagorean Theorem,

$MO = \sqrt{(MN)^{2}+(NO)^{2} }$

Substituting:

$MO= \sqrt{t^{2}+\left (\frac{1}{4}t \right)^{2} }$

$MO= \sqrt{t^{2}+ \frac {1}{16} t ^{2} }$

$MO= \sqrt{ \frac {17}{16} t ^{2} }$

Apply the Product of Radicals and Quotient of Radicals Rules:

$MO= \sqrt{ \frac {17}{16} t ^{2} }$

$= \sqrt{ \frac {17}{16} } \cdot \sqrt{t^{2}}$

$= \sqrt{ \frac {17}{16} } t$

$= \frac { \sqrt{17}}{ \sqrt{16}} \cdot t$

$= \frac { \sqrt{17}}{4} t$

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Question

Find the perimeter of a parallelogram with one side equal to 6 and the other to 5.

Answer

To solve, simply use the formula for the perimeter of a parallelogram. Thus,

If you don't remember this formula, you can also just sum up all the sides of the parallelogram.

Where s1 and s3 are the long sides and s2 and s4 are the short sides. Since the two pairs of sides are equal, you can sub in the following.

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Question

ABCD is a parallelogram. BD = 5. The angles of triangle ABD are all equal. What is the perimeter of the parallelogram?

Answer

If all of the angles in triangle ABD are equal and line BD divides the parallelogram, then all angles in triangle BDC must be equal as well.

We now have two equilateral triangles, so all sides of the triangles will be equal.

All sides therefore equal 5.

5+5+5+5 = 20

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Question

A contractor is going to re-tile a rectangular section of the kitchen floor. If the floor is 6ft x 3ft, and he is going to use square tiles with a side of 9in. How many tiles will be needed?

Answer

We have to be careful of our units. The floor is given in feet and the tile in inches. Since the floor is 6ft x 3ft. we can say it is 72in x 36in, because 12 inches equals 1 foot. If the tiles are 9in x 9in we can fit 8 tiles along the length and 4 tiles along the width. To find the total number of tiles we multiply 8 x 4 = 32. Alternately we could find the area of the floor (72 x 36, and divide by the area of the tile 9 x 9)

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Question

If the area Rectangle A is larger than Rectangle B and the sides of Rectangle A are and , what is the area of Rectangle B?

Answer

$A = \text{length}: \times :\text{width} = 10 \times 6 = 60$

$B =\frac{60}{1.25} = 48$

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Question

The front façade of a building is 100 feet tall and 40 feet wide. There are eight floors in the building, and each floor has four glass windows that are 8 feet wide and 6 feet tall along the front façade. What is the total area of the glass in the façade?

Answer

Glass Area per Window = 8 ft x 6 ft = 48 ft2

Total Number of Windows = Windows per Floor * Number of Floors = 4 * 8 = 32 windows

Total Area of Glass = Area per Window * Total Number of Windows = 48 * 32 = 1536 ft2

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Question

The rectangular bathroom floor in Michael’s house is ten feet by twelve feet. He wants to purchase square tiles that are four inches long and four inches wide to cover the bathroom floor. If each square tile costs $2.50, how much money will Michael need to spend in order to purchase enough tiles to cover his entire bathroom floor?

Answer

The dimensions for the bathroom are given in feet, but the dimensions of the tiles are given in inches; therefore, we need to convert the dimensions of the bathroom from feet to inches, because we can’t compare measurements easily unless we are using the same type of units.

Because there are twelve inches in a foot, we need to multiply the number of feet by twelve to convert from feet to inches.

10 feet = 10 x 12 inches = 120 inches

12 feet = 12 x 12 inches = 144 inches

This means that the bathroom floor is 120 inches by 144 inches. The area of Michael’s bathroom is therefore 120 x 144 in2 = 17280 in2.

Now, we need to find the area of the tiles in square inches and calculate how many tiles it would take to cover 17280 in2.

Each tile is 4 in by 4 in, so the area of each tile is 4 x 4 in2, or 16 in2.

If there are 17280 in2 to be covered, and each tile is 16 in2, then the number of tiles we need is 17280 ÷ 16, which is 1080 tiles.

The question ultimately asks us for the cost of all these tiles; therefore, we need to multiply 1080 by 2.50, which is the price of each tile.

The total cost = 1080 x 2.50 dollars = 2700 dollars.

The answer is $2700.

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Question

Ron has a fixed length of wire that he uses to make a lot. On Monday, he uses the wire to make a rectangular lot. On Tuesday, he uses the same length of wire to form a square-shaped lot. Ron notices that the square lot has slightly more area, and he determines that the difference between the areas of the two lots is sixteen square units. What is the positive difference, in units, between the length and the width of the lot on Monday?

Answer

Let’s say that the rectangular lot on Monday has a length of l and a width of w. The area of a rectangular is the product of the length and the width, so we can write the area of the lot on Monday as lw.

Next, we need to find an expression for the area of the lot on Tuesday. We are told that the lot is in the shape of a square and that it uses the same length of wire. If the length of the wire used is the same on both days, then the perimeter will have to remain the same. In other words, the perimeter of the square will equal the perimeter of the rectangle. The perimeter of a rectangle is given by 2_l_ + 2_w_.

We also know that if s is the length of a side of a square, then the perimeter is 4_s_, because each side of the square is congruent. Let’s write an equation that sets the perimeter of the rectangle and the square equal.

2_l_ + 2_w_ = 4_s_

If we divide both sides by 4 and then simplify the expression, then we can write the length of the square as follows:

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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 area 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.

A = lw = (3_x_ + 5)(2_x_) = 6_x_2 + 10_x_

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Question

Rectangle ABCD is shown in the figure above. Points A and B lie on the graph of y = 64 – _x_2 , and points C and D lie on the graph of y = _x_2 – 36. Segments AD and BC are both parallel to the y-axis. The x-coordinates of points A and B are equal to –k and k, respectively. If the value of k changes from 2 to 4, by how much will the area of rectangle ABCD increase?

Answer

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Question

Two circles of a radius of each sit inside a square with a side length of . If the circles do not overlap, what is the area outside of the circles, but within the square?

Answer

The area of a square = $\dpi{100} \small side^{2}$

The area of a circle is $\dpi{100} \small \pi r^{2}$

Area = Area of Square $\dpi{100} \small -$ 2(Area of Circle) =

$8^{2}-2\cdot \pi (2)^{2}$

$64-8\pi$

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Question

Daisy gets new carpet for her rectangluar room. Her floor is $21\ ft \times 24\ ft$ . The carpet sells for $5 per square yard. How much did she spend on her carpet?

Answer

Since $3\ ft=1\ yd$ the room measurements are 7 yards by 8 yards. The area of the floor is thus 56 square yards. It would cost $5\cdot 56=$280$ .

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Question

The length of a rectangular rug is five more than twice its width. The perimeter of the rug is 40 ft. What is the area of the rug?

Answer

For a rectangle, $P=2w+2l$ and $A=lw$ where $w$ is the width and $l$ is the length.

Let $x=width$ and $2x+5=length$ .

So the equation to solve becomes $40=2x+2(2x+5)$ or $40=6x+10$ .

Thus $x=5\ ft$ and $2x+5=15\ ft$ , so the area is $75\ ft^{2}$ .

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Question

The width of a rectangle is . The length of the rectangle is . What must be the area?

Answer

The area of a rectangle is:

$A=\textup{Length}\times \textup{ Width}$

Substitute the variables into the formula.

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