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

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.

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

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

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.

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

Add the equations and solve for y.

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

The coordinates of the point of intersection are

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.

The radius is 6 units long, choice C.

225x²+4=0

225x²=0-4

225x²=-4

x²=-4/225

x=±√-4/225

the results from a pre-test for students for the year 2000 and the year 2010 are illustrated in the box plot. what do these results tell us about how students performed on the 29 question pre-test for the two years?

if we compare only the lowest and highest scores between the two years, we might conclude that the students in 2010 did better than the students in 2010. this conclusion seems to follow since the lowest score of 8 in 2010 is greater in value than the lowest score of 6 in 2000. also, the highest score of 28 in 2010 is greater in value than the highest score of 27 in 2000.

but the box portion of the illustration gives us more detailed information. the middle bar in each box shows us that the median score of 20 in 2000 is greater in value than the median score of 17 in 2010. further, we note that the box and whiskers divide the illustration into four pieces. each of these four pieces represents the same portion of students. so, the upper half of the students in 2000 scored in the same score range as the upper one-fourth of the students in 2010, see the illustration at a score of 20.

by considering the upper one-fourth, upper half, and upper three-fourths instead of just the lowest and highest scores, we would conclude that the students as a whole did much better in 2000 than in 2010. we would conclude that as a whole the students in 2010 are less prepared than the students in 2000.

in this section, we discuss box-and-whisker plots and the five key values used in constructing a box-and-whisker plot. the key values are called a five-number summary, which consists of the minimum, first quartile, median, third quartile, and maximum.

step-by-step explanation: