a graphic artist is using a coordinate plane to design a company logo. the log has an equilateral triangle inscribed in a circle.. The lies in quadrant 1, is tangent to the x and y axis and has a radius of 10 units. one side of the triangle is parallel to the y axis and one vertex is (20,10). Write equation of the circle? What are the lengths of the sides of triangle?

Question

a graphic artist is using a coordinate plane to design a company logo. the log has an equilateral triangle inscribed in a circle.. The lies in quadrant 1, is tangent to the x and y axis and has a radius of 10 units. one side of the triangle is parallel to the y axis and one vertex is (20,10). Write equation of the circle? What are the lengths of the sides of triangle?

Answer 1

well, we know that with center at (h,k) the circle's equation is

(x-h)^2 + (y-k)^2 = 100

Since (20,10) is on the circle, at one end of the horizontal diameter, it is clear that the center of the circle is at (10,10).

(x-10)^2 + (y-10)^2 = 100

Getting the triangle's dimensions should now be no trouble.

Answer 2

To write the equation of a circle given its center and radius, we use the standard form of the equation: (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is its radius.

In this case, the circle is inscribed in the coordinate plane and its center lies in the first quadrant, which means both h and k are positive. The circle is tangent to both the x and y axes, so its center lies on the line y = 10 and x = 10.

Since the triangle is equilateral, each of its sides has the same length. We need to find that length.

One vertex of the triangle is given as (20, 10), so let's label it as A. Another vertex will be the center of the circle, which we will label as O. The third vertex will be the point where the circle intersects the y-axis, which we will label as B.

To find the length of the side of the triangle, we can use the distance formula between 2 points. Let's find the distance between A and O first:

Distance AO = √((x2 - x1)^2 + (y2 - y1)^2)
= √((10 - 20)^2 + (10 - 10)^2)
= √((-10)^2 + 0^2)
= √(100 + 0)
= 10

Since the triangle is equilateral, all sides have the same length. Therefore, the length of the side of the triangle is 10 units.

Now, let's write the equation of the circle:

Using the center O, which is (h, k) = (10, 10), and the radius r = 10, we can substitute these values into the standard form of the equation:

(x - h)^2 + (y - k)^2 = r^2
(x - 10)^2 + (y - 10)^2 = 10^2
(x - 10)^2 + (y - 10)^2 = 100

So, the equation of the circle is (x - 10)^2 + (y - 10)^2 = 100.