Find the area of a kite with diagonal lengths of and .

Explanation

Write the formula for the area of a kite.

$A= \frac{d1\cdot d2}{2}$

Plug in the given diagonals.

$A= \frac{(a+b)\cdot (2a-2b)}{2}$

Pull out a common factor of two in and simplify.

$A= \frac{(a+b)\cdot2(a-b)}{2}$

Use the FOIL method to simplify.

A convex polyhedron has twenty faces and thirty-six vertices. How many edges does it have?

Explanation

The number of vertices , edges , and faces of any convex polyhedron are related by By Euler's Formula:

Setting and solving for :

The polyhedron has 54 edges.

Find the area of a circle with a radius of .

$9\pi x^6$

$6\pi x^6$

$9\pi^2 x^6$

$6\pi^2 x^6$

$9\pi x^9$

Explanation

The area of a circle is $A=\pi r^2$ .

Substitute the radius and solve for the area.

$A=\pi (3x^3)^2 = \pi (3x^3)(3x^3) = 9\pi x^6$

The answer is: $9\pi x^6$

A circle has its origin at . The point is on the edge of the circle. What is the radius of the circle?

$r=\sqrt{74}$

$r=2\sqrt{18}$

$r=6\sqrt{2}$

$r=4\sqrt{21}$

There is not enough information to answer this question.

Explanation

The radius of the circle is equal to the hypotenuse of a right triangle with sides of lengths 5 and 7.

$r=\sqrt{74}$

This radical cannot be reduced further.

Find the area of a circle with a diameter of .

$64\pi$

$32\pi$

$16\pi$

$81\pi$

$256\pi$

Explanation

Write the formula for the area of a circle.

$A_{circle}= \pi r^2 = \frac{\pi d^2}{4}$

Substitute the diameter and solve.

$\frac{\pi (16)^2}{4} = \frac{\pi (16)(16)}{4}=\pi (16)(4)= 64\pi$

Rhombus

Solve for x and y using the rules of quadrilateral

x=6, y=9

x=9, y=6

x=2, y=4

x=6, y=10

Explanation

By using the rules of quadrilaterals we know that oppisite sides are congruent on a rhombus. Therefore, we set up an equation to solve for x. Then we will use that number and substitute it in for x and solve for y.

Right triangle 7

What is the perimeter of the triangle above?

$12+12 \sqrt{2}$

$12+ 12\sqrt{3}$

$12+ 8\sqrt{3}$

Explanation

The figure shows a right triangle. The acute angles of a right triangle have measures whose sum is $90^{\circ }$ , so

$m \angle A+ m \angle C = 90^{\circ }$

Substituting $45^{\circ }$ for $m \angle C$ :

$m \angle A+ 45^{\circ } = 90^{\circ }$

$m \angle A= 45^{\circ }$

This makes $\bigtriangleup ABC$ a 45-45-90 triangle. By the 45-45-90 Triangle Theorem, the length of leg $\overline{BC}$ is equal to that of hypotenuse $\overline{AC}$ , the length of which is 12, divided by $\sqrt{2}$ . Therefore,

$BC = \frac{AC}{\sqrt{2}} = \frac{12}{\sqrt{2}}$

Rationalize the denominator by multiplying both halves of the fraction by $\sqrt{2}$ :

$BC = \frac{12 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{12 \sqrt{2}}{2} = 6 \sqrt{2}$

By the same reasoning, $AB= 6 \sqrt{2}$ .

The perimeter of the triangle is

$P= 6 \sqrt{2} + 6 \sqrt{2} + 12 = 12+ 12 \sqrt{2}$

Example circle

Find the diameter, circumference and area of the circle above.

Diameter= 6 ft

Circumference=18.84 ft

Area= 28.27 ft2

Diameter= 6 ft

Circumference= 19 ft

Area= 30 ft2

Diameter= 9ft

Circumference=37.68 ft

Area= 28 ft2

Diameter= 3ft

Circumference=37.68 ft

Area= 28.7 ft2

Diameter= 6ft

Circumference=37.68 ft

Area= 28.27 ft2

Explanation

To find the diameter, you must know that radius is half the diameter (or the diameter is 2 times the radius.

$\small Diameter= 2(radius)$

$\small \small 2(3)= 6$

The diameter is 6ft.

To find the circumference, you must multiply the diameter (6ft) by pi.

$\small Circumference=\pi (diameter)$

$\small \small \pi(6)= 18.84$

The circumference is 18.84 ft.

To find the surface area, you must aquare the radius (3ft) and multiply by pi.

$\small Surface Area=\pi (radius^{2})$

$\small \pi(3{^2})=\pi(9)=28.27$

The surface area is 28.27 ft2.

The diameter is 6ft, the circumference is 18.84 ft, and the surface area is 28.27 ft2.

Circle a

The above circle has area $64 \pi$ . Give its equation.

$(x- 8)^{2}+ (y - 8)^2 = 64$

$(x- 8)^{2}+ (y - 8)^2 = 8$

$(x- 32)^{2}+ (y - 32)^2 = 32$

$(x- 32)^{2}+ (y - 32)^2 = 1,024$

$\textup{None of the other choices gives the correct response}$

Explanation

The equation of a circle on the coordinate plane is

$(x- h)^{2}+ (y - k)^2 = r^{2}$ ,

where are the coordinates of the center and is the radius.

The area and the radius of a circle are related by the formula

$\pi r^{2} = A$

Set $A = 64 \pi$ and solve for :

$\pi r^{2} = 64 \pi$

$\frac{ \pi r^{2} }{\pi} = \frac{64 \pi}{\pi}$

$r^{2} = 64$

$r = \sqrt{64} = 8$

As seen below, the horizontal and vertical distance from the origin to the center of the circle are both equal to this radius, and it is located in Quadrant I, so the center is :

Circle b

Setting , the equation of the circle becomes

$(x- 8)^{2}+ (y - 8)^2 = 8^{2}$

$(x- 8)^{2}+ (y - 8)^2 = 64$

On the XY plane, line segment AB has endpoints (0, a) and (b, 0). If a square is drawn with segment AB as a side, in terms of a and b what is the area of the square?