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If a circle has a radius of 5 cm, what is the length of a chord that is 6 cm awa

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Question: If a circle has a radius of 5 cm, what is the length of a chord that is 6 cm away from the center?

Options:

  1. 4 cm
  2. 6 cm
  3. 8 cm
  4. 10 cm

Correct Answer: 4 cm

Solution: 

Using the Pythagorean theorem, the length of the chord can be calculated as 2 * sqrt(5^2 - 6^2) = 2 * sqrt(25 - 36) = 2 * sqrt(-11), which is not possible. The chord cannot exist.

If a circle has a radius of 5 cm, what is the length of a chord that is 6 cm awa

Practice Questions

Q1
If a circle has a radius of 5 cm, what is the length of a chord that is 6 cm away from the center?
  1. 4 cm
  2. 6 cm
  3. 8 cm
  4. 10 cm

Questions & Step-by-Step Solutions

If a circle has a radius of 5 cm, what is the length of a chord that is 6 cm away from the center?
  • Step 1: Understand the problem. We have a circle with a radius of 5 cm and we want to find the length of a chord that is 6 cm away from the center of the circle.
  • Step 2: Recall the definition of a chord. A chord is a straight line connecting two points on the circumference of the circle.
  • Step 3: Identify the relationship between the radius, the distance from the center to the chord, and the half-length of the chord. We can use the Pythagorean theorem for this.
  • Step 4: Set up the Pythagorean theorem. The radius (5 cm) is the hypotenuse, the distance from the center to the chord (6 cm) is one leg, and half the length of the chord is the other leg.
  • Step 5: Write the equation: (half the length of the chord)^2 + (distance from center to chord)^2 = (radius)^2.
  • Step 6: Substitute the values into the equation: (half the length of the chord)^2 + 6^2 = 5^2.
  • Step 7: Simplify the equation: (half the length of the chord)^2 + 36 = 25.
  • Step 8: Rearrange the equation to find (half the length of the chord)^2: (half the length of the chord)^2 = 25 - 36.
  • Step 9: Calculate: (half the length of the chord)^2 = -11.
  • Step 10: Since we cannot have a negative value for (half the length of the chord)^2, this means that a chord cannot exist at this distance from the center.
  • Circle Geometry – Understanding the properties of circles, including chords, radii, and distances from the center.
  • Pythagorean Theorem – Applying the Pythagorean theorem to find relationships between the radius, distance from the center, and half the chord length.
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