the area of a circle inscribed in the triangle formed by the line with equation x+y=1 and the two coordinate axes can be written in the form pi(a+b(sqr.c)/d) compute the value of a+b+c+d
The radius r of an inscribed circle in a right triangle with short sides u,v and hypotenuse w is given by:
r=uv/(u+v+w)
In the given case, the x and y intercepts are 1, so u=v=1, and w=√2.
Area of circle
= πr²
= π(u²v²)/(u+v+√2)²
=π(1²1²)/(1+1+√2)²
=π/(2+√2)²
=π(2-√2)²/(2²-2)²
=π(6-4√2)/2^sup2;
=π(3-2√2)/2
=π(1.5-1*(√2)/2)
I will leave it to you to figure out a,b,c and d.
To compute the value of a+b+c+d, let's start by finding the radius of the inscribed circle.
The line with equation x+y=1 intersects the x-axis at point (1, 0) and the y-axis at point (0, 1). These two points, along with the origin (0, 0), form the vertices of the triangle.
Since the circle is inscribed in the triangle, its center will be the incenter of the triangle. The incenter is the point where the angle bisectors of the triangle intersect. In this case, the incenter lies on the perpendicular bisectors of the triangle's sides.
The triangle formed has two sides lying on the coordinate axes, so the perpendicular bisectors of these sides are the lines x = 1/2 and y = 1/2. The intersection of these two lines is the incenter of the triangle, which is the center of the inscribed circle.
Therefore, the center of the circle has coordinates (1/2, 1/2).
Now, let's find the distance from the center of the circle to any of the triangle's sides. We can choose the hypotenuse, which is the line x+y=1.
To find this distance, we can use the formula for the distance between a point (x1, y1) and a line in general form (Ax + By + C = 0):
Distance formula: d = |Ax1 + By1 + C| / sqrt(A^2 + B^2)
For the line x+y=1, we have A=1, B=1, C=-1, and the center of the circle (x1, y1) = (1/2, 1/2).
Plugging these values into the formula, we get:
d = |1(1/2) + 1(1/2) - 1| / sqrt(1^2 + 1^2)
= |1/2 + 1/2 - 1| / sqrt(1 + 1)
= |-1/2| / sqrt(2)
= 1/2sqrt(2)
= sqrt(2)/4
This distance represents the radius of the inscribed circle.
Therefore, the area of the circle can be written as:
A = πr^2 = π( (sqrt(2)/4)^2 ) = π(2/16) = π/8
Comparing this to the given form π(a + b(sqrt(c))/d) , we can conclude that:
a = 1
b = 0
c = 2
d = 8
Thus, the value of a+b+c+d is 1 + 0 + 2 + 8 = 11.