Please help, i dont understand an ounce of this

Mathematics

1 answer:

nignag [31]2 years ago

4 0

Let x be the length of each side of the nonagon. We then split up the nonagon into 9 congruent, isosceles triangles, each with base = x and height = 12. Then the area of each triangle is 1/2 • x • 12 = 6x, so the total area of the nonagon will be 9 • 6x = 54x.

To find x, we can use some facts from geometry and trigonometry.

• In any polygon, the sum of the measures of the exterior angles is 360°. So each of these exterior angles will measure 360°/9 = 40°.

• Exterior angles are supplementary to the interior angles. So each interior angle will measure 180° - 40° = 140°.

• Each of the 9 triangles are isosceles with base angles measuring half the interior angles of the nonagon, 140°/2 = 70°.

• Cut the triangle in half along the labeled inradius of the nonagon, which has length 12. In the resulting right triangle, we have

tan(70°) = 12 / (x/2)

and solving for x gives

tan(70°) = 24/x

x = 24/tan(70°)

x = 24 cot(70°) ≈ 8.7

Then the total area of the nonagon is

54x = 54 • 24 cot(70°) ≈ 471.7

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Step-by-step explanation:

The equation for a circle is $(x-h)^2+(y-k)^2=r^2$ , where $(h,k)$ is the vertex of the circle and $r$ is the radius. Immediately, since the center of the circle is given, we know what $h$ and $k$ are. $h$ is 230 and $k$ is 220.

The only thing we need to find is the radius, which will just be the distance from the center (230,220) to a point on the circumference (170,170). The distance between them can be calculated using the <u>distance formula</u>, which is really just the <u>Pythagorean Theorem</u> rearranged. The formula states that $d=\sqrt{x^2+y^2}$ , where $x$ is the change in the x-coordinates of the two points and $y$ is the change in the y-coordinates of the two points. Plug-in $x$ for 60 and $y$ for 50 to get $d=\sqrt{50^2+60^2}$ . Solve for $d$ , arriving at $\sqrt{6100}$ . Therefore, the radius of the circle is $\sqrt{6100}$ .