a piece of a broken circular gear is brought into a metal shop so that a replacement can be built. a ruler is placed across two points on the rim, and the length of the chordd is found to be 14 inches. the distance from the midpoint of this chord to the nearest point on the rim is found to be 4 inches. find the radius of the original gear.
the nearest point, the end of the chord, and the end of a diameter (opposite the point) form a right triangle
the half of the chord forms the altitude of the triangle
4 / 7 = 7 / (d - 4)
solve for the diameter (which is twice the radius)
To find the radius of the original gear, we can use the Pythagorean theorem.
Let's consider the radius of the original gear as 'r'.
From the given information, we know that the chord's length is 14 inches and the distance from the midpoint of the chord to the nearest point on the rim is 4 inches.
Let's label the endpoints of the chord as A and B, and the midpoint of the chord as M.
Now, we can draw a right triangle with AM as the base, which is half of the chord length (14/2 = 7 inches), and BM as the height, which is 4 inches.
Using the Pythagorean theorem, we have:
r^2 = AM^2 + BM^2
Plugging in the given values, we get:
r^2 = 7^2 + 4^2
r^2 = 49 + 16
r^2 = 65
Taking the square root of both sides, we find:
r ≈ √65
Therefore, the radius of the original gear is approximately √65 inches.
To find the radius of the original gear, we can use the properties of a chord and radius in a circle.
Let's denote the length of the chord as c (which is 14 inches), and the distance from the midpoint of the chord to the nearest point on the rim as d (which is 4 inches).
First, we need to find the length of the line segment (also called a perpendicular bisector) that connects the center of the circle to the midpoint of the chord. This line segment is perpendicular to the chord and bisects it.
To find this length, we can use the Pythagorean theorem. Let's call this line segment l.
Apply the theorem: l^2 = r^2 - (c/2)^2
Here, r represents the radius of the circle.
Substituting the given values, we have:
l^2 = r^2 - (14/2)^2
= r^2 - 7^2
= r^2 - 49
Next, we can use the Pythagorean theorem again to find the radius of the original gear.
Apply the theorem: r^2 = l^2 + d^2
Substituting the known values, we have:
r^2 = (4^2) + (l^2)
= 16 + l^2
= 16 + (r^2 - 49) (substituting for l^2)
Rearranging the equation, we get:
r^2 - r^2 = 16 - 49
0 = -33
This equation has no real solutions, which means there might be a mistake in the given information or the problem itself.
Please double-check the values provided in the problem and ensure their accuracy.