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

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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.

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