Other Shapes - GED Math

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. . $\angle C$ is a right angle. What percent of $\Delta ACE$ has been shaded in?

Answer

$\Delta BCD$ is a right triangle with legs ; its area is half the product of its legs, which is

$Area= \frac{1}{2} \times 3 \times 4 = 6$

$\Delta ACE$ is a right triangle with legs

and

;

its area is half the product of its legs, which is

$Area= \frac{1}{2} \times 5 \times 12 = 30$

The shaded region is the former triangle removed from the latter triangle; its area is the difference of the two: .

This region is therefore

$\frac{24}{30 } \times 100 % = 80 %$ of $\Delta ACE$ .

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. . $\angle C$ is a right angle. What is the area of the shaded region?

Answer

$\Delta BCD$ is a right triangle with legs ; its area is half the product of its legs, which is

$Area= \frac{1}{2} \times 8 \times 8 = 32$

$\Delta ACE$ is a right triangle with legs

and

;

its area is half the product of its legs, which is

$Area= \frac{1}{2} \times 12 \times 24= 144$

The shaded region is the former triangle removed from the latter triangle; its area is the difference of the two: .

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Question

The above hexagon is regular. Give its area.

Answer

A regular hexagon can be divided into six triangles, each of which can be easily proved equilateral, as seen in the diagram below:

All segments shown are congruent, and, since the diameter shown in the original diagram is 4, each sidelength is half this, or 2.

Each equilateral triangle has area

$A = \frac{s^{2} \sqrt{3}}{4} = \frac{2^{2} \sqrt{3}}{4} = \frac{4\sqrt{3}}{4} = \sqrt{3}$ .

There are six such triangles, so the total area of the hexagon is six times this, or $6\sqrt{3}$ .

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Question

Give the area of a regular hexagon with perimeter 36.

Answer

A hexagon has six sides; a regular hexagon with perimeter 36 has sidelength

$36 \div 6 = 6$ .

A regular hexagon can be divided into six triangles, each of which can be easily proved equilateral, as seen in the diagram below:

Each equilateral triangle has sidelength 6, so each has area

$A = \frac{s^{2} \sqrt{3}}{4} = \frac{6^{2} \sqrt{3}}{4} = \frac{36 \sqrt{3}}{4} = 9 \sqrt{3}$ .

The total area of the hexagon is the area of six such triangles:

$6 \cdot 9 \sqrt{3} = 54 \sqrt{3}$

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Question

Determine the area of a square with a side length of .

Answer

Write the area of a square.

Substitute the side into the formula.

$A=(2x^5)^2 = (2x^5)(2x^5) = 4x^{10}$

The answer is: $4x^{10}$

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Question

Figure NOT drawn to scale.

Refer to the above figure. Every angle shown is a right angle.

Give its area.

Answer

Examine the bottom figure, in which the bottom two sides have been connected. Note that the figure is now a rectangle cut out of a rectangle, and, since the opposite sides of a rectangle have the same length, we can fill in some of the side lengths as shown:

The figure is a 60-by-40 rectangle cut from a 100-by-100 square, so, to get the area of the figure, subtract the area of the former from that of the latter. The area of a rectangle is equal to the product of its dimensions, so the areas of the rectangle and the square are, respectively,

$40 \times 60 = 2,400$

and

$100 \times 100 = 10,000$ ,

making the area of the figure

.

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Question

A circle is inscribed in square that has a side length of , as shown by the figure below.

Find the area of the shaded region. Use $\pi=3.14$ .

Answer

Since the circle is inscribed in the square, the diameter of the circle is the same length as the length of a square.

Start by finding the area of the square.

$\text{Area of Square}=\text{side}^2$

For the given square,

$\text{Area}=144$

Now, because the diameter of the circle is the same as the length of a side of the square, we now also know that the radius of the circle must be . Next recall how to find the area of a circle.

$\text{Area of Circle}=\pi r^2$

Plug in the found radius to find the area of the circle.

$\text{Area of Circle}=\pi(6)^2=36\pi=113.04$

Now, the shaded area is the area left over when the area of the circle is subtracted from the area of the square. Thus, we can write the following equation to find the area of the shaded region.

$\text{Area of Shaded Region}=144-113.04=30.96$

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Question

Josh wants to build a circular pool in his square yard that measures feet on each side. He wants to build the pool as big as possible, then pave the rest of his yard in tile. In square feet, what is the area of the yard that will be tiled? Round your answer to the nearest tenths place.

Answer

Start by drawing out the square yard and the circular pool in a way that maximizes the area of the pool.

Notice that the diameter of the pool will be the same length as the side of the square.

Since the question asks about the area that is left over after the pool is built, we can find that area by subtracting the area of the pool from the area of the square.

Start by finding the area of the square.

$\text{Area of Square}=50^2=2500$

Next, find the area of the circular pool.

Since the diameter of the pool is , the radius of the pool must be . Recall how to find the area of a circle:

$\text{Area}=\pi r^2$

Plug in the radius of the circle.

$\text{Area}=\pi (25)^2=625\pi$

Subtract the area of the circle from that of the square.

$\text{Tiled Area}=2500-625\pi=536.5$

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Question

The above figure is a regular octagon. Give its perimeter in yards.

Answer

A regular octagon has eight sides of equal length, so multiply the length of one side by eight:

$6 \frac{1}{4} \times 8 = \frac{25}{4} \times \frac{8}{1} = \frac{200}{4} = 50$ feet.

Divide by three to get the equivalent in yards:

$50 \div 3 = 16 \frac{2}{3}$ yards.

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Question

Identify the above polygon.

Answer

A polygon with eight sides is called an octagon.

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Question

Refer to the above figure.

Which of the following is not a valid alternative name for Polygon ?

Answer

In naming a polygon, the vertices must be written in the order in which they are positioned, going either clockwise or counterclockwise. Of the four choices, only Polygon violates this convention, since and are not adjacent vertices (nor are and ).

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Question

Refer to the above figure. All angles shown are right angles.

What is the perimeter of the figure?

Answer

The figure can be viewed as the composite of rectangles. As such, we can take advantage of the fact that opposite sides of a rectangle have the same length, as follows:

Now that the missing sidelengths are known, we can add the sidelengths to find the perimeter:

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Question

Refer to the above figure.

Which of the following segments is a diagonal of Pentagon ?

Answer

A diagonal of a polygon is a segment whose endpoints are nonconsecutive vertices of the polygon. Of the four choices, only $\overline{KM}$ fits this description.

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Question

Classify the above polygon.

Answer

A polygon with eight sides is called an octagon.

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Question

Hexagon is regular. If diagonals $\overline{EO}$ and $\overline{XG}$ are constructed, which of the following classifications applies to Quadrilateral ?

I) Rectangle

II) Rhombus

III) Square

IV) Trapezoid

Answer

The figure described is below.

Since the hexagon is regular, its sides are congruent, and its angles each have measure $180 ^{\circ } \times \frac{6-2}{6} = 180 ^{\circ } \times \frac{4}{6} = 120^{\circ }$ .

Also, each of the triangles are isosceles, and their acute angles measure $\frac{1}{2} (180^{\circ }- 120^{\circ }) = 30^{\circ }$ each. This means that each of the four angles of Quadrilateral measures $120 ^{\circ }- 30^{\circ }= 90^{\circ }$ , so Quadrilateral is a rectangle. However, not all sides are congruent, so it is not a rhombus. Also, since it is a rectangle, it cannot be a trapezoid.

The correct response is I only.

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Question

Classify the above polygon.

Answer

A polygon with six sides is called a hexagon.

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Question

What is the perimeter of a semicircle with an area of $16\pi$ ?

Answer

Write the formula for the area of a semicircle.

$A = \frac{1}{2}\pi r^2$

Substitute the area.

$16\pi = \frac{1}{2}\pi r^2$

Multiply by 2, and divide by pi on both sides.

$\frac{16\pi \cdot (2)}{\pi}=\frac{ \frac{1}{2}\cdot (2)\pi r^2}{\pi}$

The equation becomes:

Square root both sides and factor the right side.

$\sqrt{r^2} =\sqrt{32}$

$r=4\sqrt{2}$

The diameter is double the radius.

$D=8\sqrt{2}$

The circumference is half the circumference of a full circle.

$C_{semi}=\frac{1}{2}\pi D = 4\pi \sqrt{2}$

The perimeter is the sum of the diameter and the half circumference.

The answer is: $8\sqrt2+ 4\pi \sqrt2$

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Question

A hexagon has a perimeter of 90in. Find the length of one side.

Answer

A hexagon has 6 equal sides. The formula to find perimeter of a hexagon is:

where a is the length of any side. Now, to find the length of one side, we will solve for a.

We know the perimeter of the hexagon is 90in. So, we will substitute and solve for a. We get

$90\text{in} = 6a$

$\frac{90\text{in}}{6} = \frac{6a}{6}$

$15\text{in} = a$

$a = 15\text{in}$

Therefore, the length of one side of the hexagon is 15in.

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Question

A hexagon has a perimeter of 138cm. Find the length of one side.

Answer

A hexagon has 6 equal sides. The formula to find perimeter of a hexagon is

where a is the length of any side. To find the length of one side, we solve for a.

Now, we know the perimeter of the hexagon is 138cm. So, we can substitute and solve for a. We get

$138\text{cm} = 6a$

$\frac{138\text{cm}}{6} = \frac{6a}{6}$

$23\text{cm} = a$

$a = 23\text{cm}$

Therefore, the length of one side of the hexagon is 23cm.

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Question

A hexagon has a perimeter of 126in. Find the length of one side.

Answer

A hexagon has 6 equal sides. The formula to find perimeter of a hexagon is:

where a is the length of any side. Now, to find the length of one side, we will solve for a.

We know the perimeter of the hexagon is 126in. So, we will substitute and solve for a. We get

$126\text{in} = 6a$

$\frac{126\text{in}}{6} = \frac{6a}{6}$

$21\text{in} = a$

$a = 21\text{in}$

Therefore, the length of one side of the hexagon is 21in.

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