Question

use the figure shown to the right to find the value of x.

\\(x \approx\\) (do not round until the final answer. then round to the nearest tenth as needed.)

Explanation:

Identify the geometric properties

In the given circle, we have a chord and a radius perpendicular to it.
A radius or line segment from the center that is perpendicular to a chord bisects the chord.
The diagram shows a right triangle formed by:

• The radius of the circle as the hypotenuse, \(r = 21\).
• The perpendicular distance from the center to the chord, which is \(4\).
• Half of the chord length, which is labeled as \(x\).

Apply the Pythagorean theorem

Using the Pythagorean theorem in the right triangle:

$$ x^2 + 4^2 = 21^2 $$

Solve for x

Isolate \(x^2\):

$$ x^2 + 16 = 441 $$

$$ x^2 = 441 - 16 $$

$$ x^2 = 425 $$

$$ x = \sqrt{425} $$

Calculate the decimal value

Simplify and calculate the square root:

$$ x = 5\sqrt{17} \approx 20.6155 $$

Rounding to the nearest tenth as requested:

$$ x \approx 20.6 $$

Answer:

Use the figure shown to the right to find the value of x.

\(x \approx\) <blank>20.6</blank> (Do not round until the final answer. Then round to the nearest tenth as needed.)