A circle at the origin whose is 2 units is 2 is inscribed in a right triangle whose base is 8 units and altitude 6 units. Find
a) The equation of the circle
b)The area and circumference of the circle
c)The area of the right triangle exluding the area covered by the circle
(a) x^2+y^2 = 4
(b) surely you know how to figure the area and circumference of a circle of radius 2.
(c) subtract that area from the area of the triangle. This is because the entire circle is within the triangle.
a) To find the equation of the circle, we need to find the coordinates of its center and its radius. Since the circle is inscribed in a right triangle, the center of the circle will be the intersection point of the perpendicular bisectors of the triangle's hypotenuse.
Step 1: Find the midpoint of the hypotenuse:
The hypotenuse of the right triangle connects the points (0,0) (the origin) and (8,6) (the base and altitude intersection point).
The midpoint of the hypotenuse is the average of these coordinates:
Midpoint of hypotenuse = ((0+8)/2, (0+6)/2) = (4, 3).
Step 2: Find the slope of the hypotenuse:
The slope of the hypotenuse can be found using the formula:
Slope = (y2 - y1) / (x2 - x1)
Slope of hypotenuse = (6-0) / (8-0) = 6/8 = 3/4.
Step 3: Find the negative reciprocal of the slope:
The perpendicular bisector of the hypotenuse will have a slope that is the negative reciprocal of the hypotenuse's slope:
Negative reciprocal of 3/4 = -4/3.
Step 4: Find the equation of the perpendicular bisector:
Using the point-slope form of a line (y - y1) = m(x - x1), where (x1, y1) is a point on the line and m is the slope, we can substitute the values:
(y-3) = (-4/3)(x-4)
3y - 9 = -4x + 16
4x + 3y = 25.
Step 5: Find the equation of the circle:
The equation of a circle with its center (h, k) and radius r is:
(x-h)^2 + (y-k)^2 = r^2.
Since the center of the circle lies on the perpendicular bisector, we can substitute the equation of the perpendicular bisector:
(3y - 25/3)^2 + (x - 4)^2 = r^2.
Since the circle is inscribed in a right triangle, the radius is actually the distance from the center of the circle to any of the triangle's vertices. In this case, it is the distance from the center to the origin, which is 2 units. Substituting this value into the equation, we get:
(3y - 25/3)^2 + (x - 4)^2 = 2^2.
Therefore, the equation of the circle is:
(3y - 25/3)^2 + (x - 4)^2 = 4.
b) To find the area and circumference of the circle, we can use the known formulae:
Area of a circle = πr^2,
Circumference of a circle = 2πr.
Substituting the radius, which is 2 units, we get:
Area of the circle = π(2^2) = 4π square units,
Circumference of the circle = 2π(2) = 4π units.
Therefore, the area of the circle is 4π square units, and the circumference is 4π units.
c) To find the area of the right triangle excluding the area covered by the circle, we need to find the area of the right triangle and subtract the area of the circle.
Area of the right triangle:
The area of a triangle can be calculated using the formula:
Area = (1/2) * base * height
Area of the right triangle = (1/2) * 8 * 6 = 24 square units.
Area covered by the circle:
Since the circle is inscribed in the triangle, its diameter is equal to the length of the hypotenuse of the triangle, which is 10 units (using the Pythagorean theorem).
Substituting the radius, which is 2 units, we get:
Area of the circle = π(2^2) = 4π square units.
Therefore, the area of the right triangle excluding the area covered by the circle = 24 - 4π square units.
To solve this problem, we can break it down into several steps.
Step 1: Finding the equation of the circle.
The center of the circle is at the origin (0,0), and its radius is 2 units. The equation of a circle with center (h, k) and radius r is given by (x-h)^2 + (y-k)^2 = r^2.
In this case, the equation of the circle is (x-0)^2 + (y-0)^2 = 2^2, which simplifies to x^2 + y^2 = 4.
Step 2: Finding the area and circumference of the circle.
The formula for the area of a circle is A = πr^2, and the formula for the circumference is C = 2πr.
Substituting the radius r = 2 into these formulas, we find:
A = π(2^2) = 4π square units,
C = 2π(2) = 4π units.
Step 3: Finding the area of the right triangle.
The formula for the area of a right triangle is A = (1/2)bh, where b is the base length and h is the altitude.
Substituting b = 8 and h = 6 into this formula, we find:
A = (1/2)(8)(6) = 24 square units.
Step 4: Finding the area of the right triangle excluding the area covered by the circle.
To find the area of the right triangle excluding the area covered by the circle, we subtract the area of the circle (A) from the area of the right triangle (A_total).
A_total - A = 24 - 4π square units.
So, the answers to the given questions are:
a) The equation of the circle is x^2 + y^2 = 4.
b) The area of the circle is 4π square units, and the circumference is 4π units.
c) The area of the right triangle excluding the area covered by the circle is 24 - 4π square units.