Quadrilaterals

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If a rectangle has a width of and a length that is double the width, what would be the area of the rectangle? Round to the nearest tenth.

Explanation

To calculate the area of a triangle, we want to multiply the length by the width. Since the length is twice that of the width, which is , we can determine length as such:

$2.3;cm\cdot2=4.6;cm$

Now that we know the values for length and width, we can calculate the area of the triangle:

$Area=2.3;cm\cdot4.6;cm=10.6;cm^2$

You recently bought a new bookshelf with a base in the shape of an isosceles trapezoid. If the small base is 2 feet, the large base is 3 feet, and the arms are 1.5 feet, what is the perimeter of the base of your new bookshelf?

Cannot be determined from the information provided.

Explanation

To find the perimeter of a bookshelf, we need to add up the lengths of the sides.

We know the two bases, we just need to add the lengths of the arms.

So, our answer is 8ft

Kite_series_2

Using the kite shown above, find the length of side

Explanation

A kite is a geometric shape that has two sets of equivalent adjacent sides.

Thus, the length of side .

Since, , must equal .

A kite has a perimeter of inches. One pair of adjacent sides of the kite have a length of inches. What is the measurement for each of the other two sides of the kite?

$\frac{7}{2}in$

$\frac{6}{5}in$

Explanation

To find the missing side of this kite, work backwards using the formula:

, where and represent the length of one side from each of the two pairs of adjacent sides.

The solution is:

$\frac{26}{2}=(6+b)$

A rectangle has an area of $96 ; cm^{2}$ . The width is four less than the length. What is the perimeter?

Explanation

For a rectangle, area is and perimeter is , where is the length and is the width.

Let = length and = width.

The area equation to solve becomes , or $x^{2}-4x-96=0$ .

To factor, find two numbers the sum to -4 and multiply to -96. -12 and 8 will work:

x2 + 8x - 12x - 96 = 0

x(x + 8) - 12(x + 8) = 0

(x - 12)(x + 8) = 0

Set each factor equal to zero and solve:

or .

Therefore the length is and the width is , giving a perimeter of .

A square has perimeter 1.

True or false: The area of the square is $\frac{1}{2}$ .

False

True

Explanation

All four sides of a square have the same length, so the common sidelength is one fourth of the perimeter. The perimeter of the given square is 1, so the length of each side is $\frac{1}{4}$ .

The area of a square is equal to the square of the length of a side, so the area of this square is

$A =\left ( \frac{1}{4} \right )^{2} = \frac{1^{2} }{4^{2} } = \frac{1}{16}$ .

Find the area of the parallelogram.

Explanation

Recall how to find the area of a parallelogram:

$\text{Area}=\text{base}\times\text{height}$

From the given parallelogram, we will need to use the Pythagorean Theorem to find the length of the height of the parallelogram.

$\text{Hypotenuse}^2=\text{Triangle base}^2+\text{height}^2$

$\text{height}^2=\text{Hypotenuse}^2-\text{Triangle base}^2$

$\text{height}=\sqrt{\text{Hypotenuse}^2-\text{Triangle base}^2}$

Plug in the given values to find the length of the height.

$\text{height}=\sqrt{5^2-2^2}=\sqrt{25-4}=\sqrt{21}$

Now, use the height to find the area of the parallelogram.

$\text{Area}=12\times\sqrt{21}=54.99$

Remember to round to places after the decimal.

In the figure, the area of the parallelogram is . Find the length of the base.

Cannot be determined

Explanation

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=(\text{base})(\text{height})$

Now, substitute in the area, base, and height values that are given by the question.

Expand this equation.

Now factor this equation.

Solve for .

Since the values of lengths of shapes can only be positive, we know that the base of the parallelogram must be .

A parallelogram has a height of and an area of . What is the length of the base of the parallelogram?

Explanation

To find the missing side of this parallelgram apply the formula: $A=base\times height$

Thus, the solution is:

$168=12\times base$

$base=\frac{168}{12}=14$

If the height of the parallelogram is half of the length of the rectangle, then find the area of the shaded region in the figure.

Explanation

In order to find the area of the shaded region, we must first find the areas of the rectangle and parallelogram.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the given length and height to find the area of the rectangle.

$\text{Area of Rectangle}=8 \times 14$

$\text{Area of Rectangle}=112$

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=\text{base}\times\text{height}$

We need to find the height of the parallelogram. From the question, we know the following relationship:

$\text{height}=\frac{\text{length}}{2}$

Substitute in the length of the rectangle to find the height of the parallelogram.

$\text{height}=\frac{8}{2}$

$\text{height}=4$

Now, substitute in the height and the given length of the base to find the area of the parallelogram.

$\text{Area of Parallelogram}=10 \times 4$

$\text{Area of Parallelogram}=40$

Now, we are ready to find the area of the shaded region by subtracting the area of the parallelogram from the area of the rectangle.

$\text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$

$\text{Area of Shaded Region}=112-40$

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