Quadrilaterals - Math

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Question

The area of square R is 12 times the area of square T. If the area of square R is 48, what is the length of one side of square T?

Answer

We start by dividing the area of square R (48) by 12, to come up with the area of square T, 4. Then take the square root of the area to get the length of one side, giving us 2.

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Question

When the side of a certain square is increased by 2 inches, the area of the resulting square is 64 sq. inches greater than the original square. What is the length of the side of the original square, in inches?

Answer

Let x represent the length of the original square in inches. Thus the area of the original square is x2. Two inches are added to x, which is represented by x+2. The area of the resulting square is (x+2)2. We are given that the new square is 64 sq. inches greater than the original. Therefore we can write the algebraic expression:

x2 + 64 = (x+2)2

FOIL the right side of the equation.

x2 + 64 = x2 + 4x + 4

Subtract x2 from both sides and then continue with the alegbra.

64 = 4x + 4

64 = 4(x + 1)

16 = x + 1

15 = x

Therefore, the length of the original square is 15 inches.

If you plug in the answer choices, you would need to add 2 inches to the value of the answer choice and then take the difference of two squares. The choice with 15 would be correct because 172 -152 = 64.

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Question

If the area of a square is 25 inches squared, what is the perimeter?

Answer

The area of a square is equal to length times width or length squared (since length and width are equal on a square). Therefore, the length of one side is $l = \sqrt{25in^{2}}$ or $l=5 in.$ The perimeter of a square is the sum of the length of all 4 sides or $4 \times 5 in. =20 in.$

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Question

A square is inscribed inside a circle, as illustrated above. The radius of the circle is $\frac{3\sqrt{2}}{2}$ units. If all of the square's diagonals pass through the circle's center, what is the area of the square?

Answer

Given that the square's diagonals pass through the circle's center, those diagonals must each form a diameter of the circle. The circle's diameter is twice its radius, i.e. $2 \times \frac{3\sqrt{2}}{2}$ , which is $3\sqrt{2}$ . Since this diameter (i.e., the square's diagonal) is the hypotenuse of a right triangle formed by two sides of the square, the length of one of the square's sides can be calculated with the Pythagorean Theorem. replace and because the sides of the square must be equal in length. Since the objective is to solve for the square's area, solve for since one side squared will be the square's area.

$s^2 + s^2 = (3\sqrt{2})^2$

$\rightarrow 2s^2 = 18$

$\rightarrow s^2 = 9$ units squared

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Question

What is the area of this regular trapezoid?

Answer

To solve this question, you must divide the trapezoid into a rectangle and two right triangles. Using the Pythagorean Theorem, you would calculate the height of the triangle which is 4. The dimensions of the rectangle are 5 and 4, hence the area will be 20. The base of the triangle is 3 and the height of the triangle is 4. The area of one triangle is 6. Hence the total area will be 20+6+6=32. If you forget to split the shape into a rectangle and TWO triangles, or if you add the dimensions of the trapezoid, you could arrive at 26 as your answer.

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Question

What is the area of the trapezoid above if a = 2, b = 6, and h = 4?

Answer

Area of a Trapezoid = ½(a+b)h

= ½ (2+6) 4

= ½ (8) * 4

= 4 * 4 = 16

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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

This figure is an isosceles trapezoid with bases of 6 in and 18 in and a side of 10 in.

What is the area of the isoceles trapezoid?

Answer

In order to find the area of an isoceles trapezoid, you must average the bases and multiply by the height.

The average of the bases is straight forward:

$\frac{6+18}{2}=12$

In order to find the height, you must draw an altitude. This creates a right triangle in which one of the legs is also the height of the trapezoid. You may recognize the Pythagorean triple (6-8-10) and easily identify the height as 8. Otherwise, use $a^{2}+b^{2}=c^{2}$ .

$36+b^{2}=100$

$b^{2}=64$

Multiply the average of the bases (12) by the height (8) to get an area of 96.

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Question

Find the area of the following trapezoid:

Answer

The formula for the area of a trapezoid is:

$A = \frac{1}{2}(b_{1}+b_{2})(h)$

Where is the length of one base, is the length of the other base, and is the height.

To find the height of the trapezoid, use a Pythagorean triple:

Plugging in our values, we get:

$A = \frac{1}{2}(b_{1}+b_{2})(h)$

$A = \frac{1}{2}(16m+26m)(12m)=252m^2$

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Question

Find the area of the following trapezoid:

Answer

Use the formula for triangles in order to find the length of the bottom base and the height.

The formula is:

$a-a\sqrt{3}-2a$

Where is the length of the side opposite the $\measuredangle 30$ .

Beginning with the side, if we were to create a triangle, the length of the base is $6\sqrt{3}m$ , and the height is .

Creating another triangle on the left, we find the height is , the length of the base is $2\sqrt{3}m$ , and the side is $4\sqrt{3}m$ .

The formula for the area of a trapezoid is:

$A = \frac{1}{2}(b_{1}+b_{2})(h)$

Where is the length of one base, is the length of the other base, and is the height.

Plugging in our values, we get:

$A = \frac{1}{2}(b_{1}+b_{2})(h)$

$A = \frac{1}{2}(16m+16m+6\sqrt{3}m+2\sqrt{3}m)(6m)$

$A = \frac{1}{2}(32m+8\sqrt{3}m)(6m)$

$A = (32m+8\sqrt{3}m)(3m)=96+24\sqrt{3}m^2$

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Question

Determine the area of the following trapezoid:

Answer

The formula for the area of a trapezoid is:

$A=\frac{1}{2}(b_{1}+b_{2})(h)$ ,

where is the length of one base, is the length of another base, and is the length of the height.

Plugging in our values, we get:

$A=\frac{1}{2}(7m+5m)(4m)=24m^2$

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Question

Find the area of the following trapezoid:

Answer

The formula for the area of a trapezoid is:

$A=\frac{1}{2}(b_{1}+b_{2})(h)$ ,

where is the length of one base, is the length of another base, and is the length of the height.

Use the Pythagorean Theorem to find the height of the trapezoid:

$B=2\sqrt{91}m$

Plugging in our values, we get:

$A=\frac{1}{2}(18m+30m)(2\sqrt{91}m)=48\sqrt{91}m^2$

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Question

Find the area of the following trapezoid:

Answer

The formula for the area of a trapezoid is

.

Use the Pythagorean Theorem to find the length of the height:

$(5\ m)^2 + B^2 = (13\ m)^2$

$B = 12\ m$

Plugging in our values, we get:

$A = (12\ m)(12\ m)$

$A = 144\ m^2$

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Question

Find the area of the following trapezoid:

Answer

The formula for the area of a trapezoid is

$A = \frac{1}{2}(b_1+b_2)(h)$ ,

where is the base of the trapezoid and is the height of the trapezoid.

Use the formula for a triangle to find the length of the base and height:

$a-2a-a\sqrt{3}$

$5\ m-10\ m-5\sqrt{3}\ m$

Use the formula for a triangle to find the length of the base and height:

$5\ m-5\ m-5\sqrt{2}\ m$

Plugging in our values, we get:

$A = \frac{1}{2}(10\ m+5\ m+10\ m+5\sqrt{3}\ m)(5\ m)$

$A = \frac{125+25\sqrt{3}\ m^2}{2}$

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Question

Quadrilateral ABCD contains four ninety-degree angles. Which of the following must be true?

I. Quadrilateral ABCD is a rectangle.

II. Quadrilateral ABCD is a rhombus.

III. Quadrilateral ABCD is a square.

Answer

Quadrilateral ABCD has four ninety-degree angles, which means that it has four right angles because every right angle measures ninety degrees. If a quadrilateral has four right angles, then it must be a rectangle by the definition of a rectangle. This means statement I is definitely true.

However, just because ABCD has four right angles doesn't mean that it is a rhombus. In order for a quadrilateral to be considered a rhombus, it must have four congruent sides. It's possible to have a rectangle whose sides are not all congruent. For example, if a rectangle has a width of 4 meters and a length of 8 meters, then not all of the sides of the rectangle would be congruent. In fact, in a rectangle, only opposite sides need be congruent. This means that ABCD is not necessarily a rhombus, and statement II does not have to be true.

A square is defined as a rhombus with four right angles. In a square, all of the sides must be congruent. In other words, a square is both a rectangle and a rhombus. However, we already established that ABCD doesn't have to be a rhombus. This means that ABCD need not be a square, because, as we said previously, not all of its sides must be congruent. Therefore, statement III isn't necessarily true either.

The only statement that has to be true is statement I.

The answer is I only.

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Question

A quadrilateral with congruent sides is a .

Answer

A rhombus is any quadrilateral with four congruent sides. A square is a quadrilateral with four congruent sides and four right angles. A square is also a rhombus.

Rectangles will have two pairs of congruent sides, but not all four sides will be congruent.

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Question

A quadrilateral ABCD has diagonals that are perpendicular bisectors of one another. Which of the following classifications must apply to quadrilateral ABCD?

I. parallelogram

II. rhombus

III. square

Answer

If the diagonals of a quadrilateral are perpendicular bisectors of one another, then the quadrilateral must be a rhombus, but not necessarily a square. Since all rhombi are also parallelograms, quadrilateral ABCD must be both a rhombus and parallelogram.

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Question

Mark is making a plan to build a rectangular garden. He has 160 feet of fence to form the outside border of the garden. He wants the dimensions to look like the plan outlined below:

What is the area of the garden, rounded to the nearest square foot?

Answer

Perimeter: Sum of the sides:

4x + 4x + 2x+8 +2x+8 = 160

12x + 6 = 160

12x = 154

x = $\frac{77}{6}$

Therefore, the short side of the rectangle is going to be:

$\dpi{100} \dpi{100} 2\cdot \frac{77}{6}+3=\frac{172}{6}=\frac{86}{3}$

And the long side is going to be:

$\dpi{100} 4\cdot \frac{77}{6}=\frac{308}{6}=\frac{154}{3}$

The area of the rectangle is going to be as follows:

Area = lw

$\dpi{100} Area = \frac{86}{3}\cdot \frac{154}{3}=\frac{13,244}{9}=1471.55\approx 1472\ ft^{2}$

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Question

Erin is getting ready to plant her tulip garden. She wants to plant two tulips per square foot of garden. If her rectangular garden is enclosed by 24 feet of fencing, and the length of the fence is twice as long as its width, how many tulips will Erin plant?

Answer

We know that the following represents the formula for the perimeter of a rectangle:

In this particular case, we are told that the length of the fence is twice as long as the width. We can write this as the following expression:

Use this information to substitute in a variable for the length that matches the variable for width in our perimeter equation.

We also know that the length is two times the width; therefore, we can write the following:

$l = 2\times4$

The area of a rectangle is found by using this formula:

$A=8\times4$

The area of the garden is 32 square feet. Erin will plant two tulips per square foot; thus, she will plant 64 tulips.

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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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