in
a circle of radius 17cm, two parallel chords are drawn on opposite
sides of a diameter. the distance between the chords is 23cm. if the
length of one chords is 16cm, find the length of the other.
Assuming the chords are parallel, think of 8-15-17 right triangles.
To find the length of the other chord, we can use the property of circles that states that chords that are equidistant from the center of the circle are congruent.
Here's how we can solve the problem step by step:
Step 1: Draw a diagram that represents the given information.
- Draw a circle with a radius of 17 cm.
- Draw a diameter that passes through the center of the circle.
- Draw two parallel chords on opposite sides of the diameter.
- Label the distance between the chords as 23 cm.
- Label the length of one chord as 16 cm.
Step 2: Use the fact that the chords are parallel to the diameter and equidistant from the center of the circle to determine the distance from each chord to the center of the circle.
- The distance from the center of the circle to the chord is equal to half the distance between the chords, which is 23 cm divided by 2, which is 11.5 cm.
Step 3: Use the Pythagorean theorem to find the length of the other chord.
- Let x represent the length of the other chord.
- In the triangle formed by the radius, one-half of the chord, and the distance from the chord to the center of the circle, we have a right triangle.
- The hypotenuse of this right triangle is the radius of the circle, which is 17 cm.
- One of the legs of the right triangle is half the length of the chord, which is 16 cm divided by 2, which is 8 cm.
- The other leg of the right triangle is the distance from the chord to the center of the circle, which is 11.5 cm.
- Using the Pythagorean theorem, we can write the equation: (8 cm)^2 + (11.5 cm)^2 = (17 cm)^2
- Simplifying the equation, we get: 64 cm^2 + 132.25 cm^2 = 289 cm^2
- Combining like terms, we get: 196.25 cm^2 = 289 cm^2
- Subtracting 196.25 cm^2 from both sides, we get: 92.75 cm^2 = 0 cm^2
- Taking the square root of both sides, we get: x = 0 cm (since the left side is 0, the length of the other chord in this case would be 0 cm).
Therefore, the length of the other chord is 0 cm.
Note: It's important to carefully double-check the given information and calculations to ensure accuracy, as the result of 0 cm may indicate a math error or a problematic setup in the problem statement.