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In a certain quadrilateral, three of the angles are $65^{\circ}$ , $120^{\circ}$ , and $34^{\circ}$ . What is the measure of the fourth angle?

$141^{\circ}$

$139^{\circ}$

$219^{\circ}$

$41^{\circ}$

$29^{\circ}$

Explanation

A quadrilateral has four angles totalling $360^{\circ}$ . So, first add up the three angles given. The sum is $219^{\circ}$ . Then, subtract that from 360. This gives you the missing angle, which is $141^{\circ}$ .

Circle_graph_area3

100_π_

50_π_

25_π_

10_π_

20_π_

Explanation

Circle_graph_area2

What is the area of the figure below?

Explanation

To find the area of the figure above, we need to slip the figure into two rectangles.

11.5

Using our area formula, $A=l\times w$ , we can solve for the area of both of our rectangles

$A=7\times 3$ $A=6\times 3$

To find our final answer, we need to add the areas together.

The lengths of the sides of ten squares form an arithmetic sequence. One side of the smallest square measures sixty centimeters; one side of the second-smallest square measures one meter.

Give the area of the largest square, rounded to the nearest square meter.

18 square meters

16 square meters

20 square meters

22 square meters

24 square meters

Explanation

Let $a_{n}$ be the lengths of the sides of the squares in meters. $a_{1} = 60 \div 100 = 0.6$ and $a_{2} = 1$ , so their common difference is

The arithmetic sequence formula is

$a_{n} = a_{1} + (n-1)d$

The length of a side of the largest square - square 10 - can be found by substituting $a_{1} = 0.6, n= 10, d = 0.4$ :

$a_{10} =0.6+ (10-1) 0.4 = 0.6 + 9 (0.4) = 0.6 + 3.6 = 4.2$

The largest square has sides of length 4.2 meters, so its area is the square of this, or $A = 4.2^{2} = 17.64$ square meters.

Of the choices, 18 square meters is closest.

What is the area of a rectangle with a length of and a width of ?

Explanation

The area of a rectangle is the length times the width:

Plug in our given values and solve:

Find the area of the shaded region:

Screen_shot_2014-02-27_at_6.35.30_pm

$108\pi-81\sqrt{3}m^2$

$108\pi-81\sqrt{2}m^2$

$81\pi-81\sqrt{3}m^2$

$108\pi-108\sqrt{3}m^2$

$81\pi-108\sqrt{3}m^2$

Explanation

To find the area of the shaded region, you must subtract the area of the triangle from the area of the sector.

The formula for the shaded area is:

$A=A_{sector}-A_{triangle}$

$A = \pi(r^2)(part)-\frac{1}{2}(b)(h)$ ,

where is the radius of the circle, is the fraction of the sector, is the base of the triangle, and is the height of the triangle.

In order to the find the base and height of the triangle, use the formula for a triangle:

$a-a\sqrt{3}-2a$ , where is the side opposite the $\measuredangle30$ .

Plugging in our final values, we get:

$A = \pi(18m^2)(\frac{120}{360})-\frac{1}{2}(18\sqrt{3}m)(9m)$

$A = 108\pi-81\sqrt{3}m^2$

Find the area of a rectangle with a length of 16cm and a width that is a quarter of the length.

$64\text{cm}^2$

$128\text{cm}^2$

$31\text{cm}^2$

$32\text{cm}^2$

$62\text{cm}^2$

Explanation

To find the area of a rectangle, we will use the following formula:

$A = l \cdot w$

where l is the length and w is the width of the rectangle.

Now, we know the length of the rectangle is 16cm. We also know the width of the rectangle is a quarter of the length. To find the width, we will divide 16 by 4. Therefore, the width the 4cm.

Knowing this, we will substitute into the formula. We get

$A = 16\text{cm} \cdot 4\text{cm}$

$A = 64\text{cm}^2$

What is the area of the figure below?

Explanation

To find the area of the figure above, we need to slip the figure into two rectangles.

12.5

Using our area formula, $A=l\times w$ , we can solve for the area of both of our rectangles

$A=6\times 8$ $A=7\times 5$

To find our final answer, we need to add the areas together.

Trapezoid 1

The above figure depicts Trapezoid with midsegment $\overline{MN }$ . Express in terms of .

Explanation

The midsegment of a trapezoid has as its length half the sum of the lengths of the bases, which here are $\overline{TR}$ and $\overline{AP }$ :

$MN = \frac{1}{2} (TR + AP )$

$20 = \frac{1}{2} (x+ y)$

$20 \cdot 2 = \frac{1}{2} (x+ y) \cdot 2$

The correct choice is .

The perimeter of a square is one yard. Which is the greater quantity?

(a) The area of the square

(b) $\frac{1}{2}$ square foot

(a) is greater.

(b) is greater.

(a) and (b) are equal.

It is impossible to tell form the information given.

Explanation

One yard is equal to three feet, so the length of one side of a square with this perimeter is $\frac{3}{4}$ feet. The area of the square is $\frac{3}{4}\times \frac{3}{4} = \frac{9}{16}$ square feet. $\frac{9}{16} > \frac{1}{2}$ , making (a) greater.

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