If a chord 10 inches long is 5 inches from the center of a circle, find the radius of the circle.
To find the radius of the circle, you can use the properties of a circle's chord and radius relationship.
Step 1: Recall the relationship between the chord, radius, and perpendicular bisector. For a chord in a circle, the perpendicular bisector of the chord passes through the center of the circle.
Step 2: Divide the chord into two equal parts, which will result in two right triangles.
Step 3: Label one of the halves of the chord as "a," and label the distance from the center of the circle to the midpoint of the chord as "b." In this case, "a" equals 5 inches.
Step 4: Apply the Pythagorean theorem to one of the right triangles. The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
a² = b² + c²
Plug in the known values:
(5 inches)² = b² + r²
Simplify the equation:
25 = b² + r²
Step 5: Substitute the values from the information given. The chord length is 10 inches, and half of the chord length (b) is 5 inches.
25 = 5² + r²
25 = 25 + r²
Step 6: Solve for r².
Subtract 25 from both sides of the equation:
r² = 0
Step 7: Take the square root of both sides to find the value of r.
r = √0
Since the square root of zero is zero, the radius of the circle is 0 inches.
Therefore, the radius of the circle is 0 inches.
To find the radius of the circle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the two other sides.
In this case, the chord of the circle forms a right triangle with the radius of the circle and a line segment perpendicular to the chord that bisects it. So, let's use this information to solve the problem step by step:
1. Draw a diagram or visualize the circle, chord, and the perpendicular segment bisecting the chord.
2. Label the radius of the circle as 'r' (the unknown we want to find) and the distance from the center of the circle to the chord as 'd' (which is given as 5 inches).
3. The length of the chord is 10 inches. Since the perpendicular segment bisects the chord, we can divide it into two segments, each 5 inches long.
4. We now have a right triangle where the hypotenuse is the radius 'r' of the circle, and the two other sides are 'd' and '5 inches.'
According to the Pythagorean theorem, we can write the equation as follows:
r^2 = d^2 + (1/2 * chord length)^2
Plugging in the values we know,
r^2 = 5^2 + (1/2 * 10)^2
r^2 = 25 + 25
r^2 = 50
To find the radius 'r,' we can take the square root of both sides:
r = √50
r ≈ 7.07 inches
So, the radius of the circle is approximately 7.07 inches.