A quarter-circle with radius 5 is drawn. A circle is drawn inside the sector, which is tangent to the sides of the sector. Find the radius of the inscribed circle.
I assume the inscribed circle is also tangent to the larger circle.
Draw a diagram. If the inscribed circle has radius r, then
r^2 + r^2 = (5-r)^2
r = 5(√2-1)
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www.wolframalpha.com/input/?i=plot+x%5E2%2By%5E2+%3D+25%2C+%28x-%285%28%E2%88%9A2-1%29%29%29%5E2%2B+%28y-%285%28%E2%88%9A2-1%29%29%29%5E2+%3D+%285%28%E2%88%9A2-1%29%29%5E2%2C+x%3D0%2C+y%3D0
To find the radius of the inscribed circle, we can use the fact that in a circle, the radius that is perpendicular to a tangent line is bisected by the tangent.
First, let's start by finding the radius of the quarter-circle. Since the quarter-circle has a radius of 5, the length of the curved part is 1/4th of the circumference of a full circle with a radius of 5, which is 2πr. Thus, the length of the curved part is 1/4 * 2π(5) = 5π/2 units.
Next, we draw the tangent lines that touch the sides of the quarter-circle sector. These tangent lines divide the quarter-circle into a triangle and a circular segment.
The triangular part can be split into two right-angled triangles. The hypotenuse of each triangle is the radius of the quarter-circle, which is 5 units. Thus, the length of each side of the triangle is 5/√2 units (using the Pythagorean theorem).
Now, let's consider the circular segment. The circular segment is a part of the circle and is bound by the two tangent lines and the arc. The arc that bounds the segment is part of the quarter-circle and has a length of 5π/2. The radius of the segment can be found by using the formula for the length of an arc: the length of the arc is equal to the radius times the central angle (in radians). In this case, the central angle is 90 degrees (π/2 radians), so the radius of the segment is (5π/2) / (π/2) = 5 units.
Now we have a right-angled triangle with one side of 5 units and a hypotenuse of (5+5/√2) units. The other side of the triangle represents the radius of the inscribed circle. We can use the Pythagorean theorem to find the length of this side:
(radius of the inscribed circle)^2 + (5/√2)^2 = (5+5/√2)^2
Simplifying this equation, we can solve for the radius of the inscribed circle.