Mathematics, 01.12.2019 12:31 kinglightskin2k

The three sides of a triangle are tangent to a unique circle called the incircle. on the coordinate plane, the incircle of abc has its center at the origin. the lines whose equations are x+4y=6s(qrt17), 2x+(sqrt5)y=-18, and 2(sqrt2)x=y+18 contain the sides of abc. what is the length of the radius of the incircle?

sqrt= square root

a. 2 units
b. 3 units
c. 6 units
d. 8 units

Answers: 3

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Answer from: diyagawande

We are looking for the length of a radius.

If we can find the coordinates of the two endpoints of a radius, then we can find its length.

A tangent to a circle intersects the circle at a single point.

A radius of the circle drawn to the point of tangency is perpendicular to the tangent.

We know the center of the circle is the origin. That means we know one endpoint of every radius.

We use one side of the triangle which is given as an equation.

We find its slope. Then using that slope and the coordinates of the origin,

we can find the equation of the radius that is perpendicular to that side.

We then solve the equations simultaneously to find the coordinates of the point of tangency. The point of tangency is the other endpopint of the radius.

Knowing two endpoints of a radius, we can find its length. That will give us the answer since we are looking for the length of the radius.

Let's use the first given equation.

Find its slope.

$x + 4y = 6 \sqrt{17}$

$4y = -x + 6 \sqrt{17}$

$y = -\dfrac{1}{4}x + \dfrac{3\sqrt{17}}{2}$

The slope of the side of the triangle is -1/4.

The slope of the radius perpendicular to that side is 4.

The equation of the line that contains the radius is

y - y_1 = m(x - x_1)

We are told the circle is centered at the origin, so one point on the line containing the radius is (0, 0).

y - 0 = 4(x - 0)

y = 4x

The line containing the radius is y = 4x.

Now we use the equation of the line containing the side of the triangle and the equation of the line containing the radius as a system of equations to find their point of intersection. That point is the other endpoint of the radius.

$x + 4y = 6 \sqrt{17}$

y = 4x

Multiply the first equation by 4 and subtract the x-term from both sides.

$16y = -4x + 24 \sqrt{17}$

y = 4x

Add the equations and solve for y.

$17y = 24 \sqrt{17}$

$y = \dfrac{24 \sqrt{17}}{17} = \dfrac{24}{\sqrt{17}}$

Now replace y with the value we found, and solve for x in the second equation.

$\dfrac{24 \sqrt{17}}{17} = 4x$

$\dfrac{6 \sqrt{17}}{17} = x$

$x = \dfrac{6 \sqrt{17}}{17} = \dfrac{6}{\sqrt{17}}$

The coordinates of the point of intersection are

$(\dfrac{6}{\sqrt{17}}, \dfrac{24}{\sqrt{17}})$

This point of intersection is one endpoint of the radius.

The other endpoint of the radius is the origin.

Now we need to find the the length of the radius.

$r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

$r = \sqrt{(\dfrac{6}{\sqrt{17}} - 0)^2 + (\dfrac{24}{\sqrt{17}} - 0)^2}$

$r = \sqrt{\dfrac{36}{17} + \dfrac{576}{17}}$

$r = \sqrt{\dfrac{612}{17}}$

$r = \sqrt{36}$

r = 6

The radius is 6 units long, choice C.

Answer from: Quest

Answer: 10pi

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.

Find the length of ac express your answer in terms of pi

Answer from: Quest

The answer is lol i dont know

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