Quadrilaterals

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Questions 1 - 10

Find the area of a rhombus if the both diagonals have a length of $\sqrt{x+2}$ .

$\frac{x+2}{2}$

$\frac{\sqrt{2x+4}}{2}$

$\sqrt{2x+4}$

Explanation

Write the formula for the area of a rhombus.

$A=\frac{d_1 \cdot d_2}{2}$

Since both diagonals are equal, . Plug in the diagonals and reduce.

$A=\frac{(\sqrt x+2)(\sqrt x+2)}{2}=\frac{\sqrt{x+2}^2}{2}= \frac{x+2}{2}$

Rhombus_1

The above figure shows a rhombus . Give its area.

$40 \sqrt{11}$

$40\sqrt{61}$

Explanation

Construct the other diagonal of the rhombus, which, along with the first one, form a pair of mutual perpendicular bisectors.

Rhombus_1

By the Pythagorean Theorem,

$n = \sqrt{12^{2}- 10^{2}} =\sqrt{144- 100} = \sqrt{44}= \sqrt{4}\cdot \sqrt{11} = 2 \sqrt{11}$

The rhombus can be seen as the composite of four congruent right triangles, each with legs 10 and $2 \sqrt{11}$ , so the area of the rhombus is

$A = 4 \cdot \frac{1}{2}\cdot 10 \cdot 2\sqrt{11} =40 \sqrt{11}$ .

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:

$168=12\times base$

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

Find the area of the rhombus.

Explanation

Recall that one of the ways to find the area of a rhombus is with the following formula:

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

Now, since all four sides in the rhombus are the same, we know from the given side value what the length of the base will be. In order to find the length of the height, we will need to use sine.

$\sin x=\frac{\text{height}}{\text{side}}$ , where is the given angle.

$\text{height}=(\text{side})(\sin x)$

Now, plug this into the equation for the area to get the following equation:

$\text{Area}=(side)^2 \sin x$

Plug in the given side length and angle values to find the area.

$\text{Area}=(5)^2\sin 33=13.62$

Make sure to round to places after the decimal.

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 .

Which of the following shapes is a trapezoid?

Shapes

Explanation

A trapezoid is a four-sided shape with straight sides that has a pair of opposite parallel sides. The other sides may or may not be parallel. A square and a rectangle are both considered trapezoids.

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.

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

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

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:

$168=12\times base$

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

Which of the following shapes is a trapezoid?

Shapes

Explanation

A trapezoid is a four-sided shape with straight sides that has a pair of opposite parallel sides. The other sides may or may not be parallel. A square and a rectangle are both considered trapezoids.

Find the area of the figure.

Explanation

From the figure, you should notice that it is made up of a right triangle and a trapezoid. The lower base of the trapezoid is also the hypotenuse of the right triangle.

First, find the length of the hypotenuse of the right triangle using the Pythagorean Theorem.

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