A plane intersects a sphere that has a radius of 13cm. The distance from the center of the sphere to the closest point on the plane is 5 cm. What is the radius of the circle that is the intersection of the sphere and the plane?
A. 8cm
B. 10cm
C. 12cm
D. 13cm
Could please shed some light on how this problem is worked out.
draw a cross-section diagram, that is,
draw a circle with radius 13
draw a chord which is 5 cm from the centre.
Draw a perpendicular from the circle to the plane.
You have 2 congruent right-angled triangles with hypotenuse 13 and side 5,
let the other side be x
x^2 + 5^2 = 13^2
x^2= 144
x = 12
Since the chord is diameter of the circle on the plane, the radius of that circle is 12
To solve this problem, we can use the Pythagorean theorem. Let's denote the radius of the circle (intersection of the sphere and plane) as r.
The plane intersects the sphere at two points: one is closest to the center of the sphere and the other is farthest from the center. The distance from the center of the sphere to the closest point on the plane is given as 5 cm.
Now, consider a right triangle formed by the center of the sphere, the closest point on the plane, and the center of the circle. The hypotenuse of this right triangle is the radius of the sphere (13 cm) and one of the legs is the distance from the center of the sphere to the closest point on the plane (5 cm).
Using the Pythagorean theorem, we can write the equation:
r^2 = 13^2 - 5^2
Simplifying this equation, we get:
r^2 = 169 - 25
r^2 = 144
Taking the square root of both sides, we find:
r = 12 cm
Therefore, the radius of the circle that is the intersection of the sphere and the plane is 12 cm. So, the correct answer is option C: 12cm.
To solve this problem, we need to use the concept of distance between a point and a plane. The center of the sphere is a point, and the plane is defined by some equation.
Given that the distance from the center of the sphere to the closest point on the plane is 5 cm, we can conclude that the radius of the circle of intersection will be 5 cm.
This is because the radius forms a right angle with the point on the circle closest to the plane. So, the radius acts as a height of a right triangle, and the distance is the hypotenuse. The other side of the right triangle is the perpendicular distance from the center of the sphere to the plane, which is given as 5 cm.
Therefore, the radius of the circle that is the intersection of the sphere and the plane is also 5 cm. This means the correct option is D. 13 cm is the radius of the sphere, not the circle of intersection.