Question

find the value of x to the nearest tenth.
8.
9.
10.

Explanation:

8. Step1: Apply the Pythagorean theorem

In a circle, if two chords are related as in the figure, we can use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). Here, \(c = 18\), \(a = 14\), and we need to find \(x\).

$$x=\sqrt{18^{2}-14^{2}}$$

8. Step2: Calculate the value

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9. Step1: Use the property of chords and the Pythagorean theorem

Half - length of the chord \(l=\frac{10}{2} = 5\). Let the radius be \(x\), and the distance from the center to the chord is \(4.2\). By the Pythagorean theorem \(x^{2}=4.2^{2}+5^{2}\)

$$x=\sqrt{17.64 + 25}$$

9. Step2: Calculate the value

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10. Step1: Apply the Pythagorean theorem

Let the radius \(r = 15\). First, find the length of the other segment of the chord using the Pythagorean theorem for the right - triangle with hypotenuse \(15\) and one leg \(8\). Let the half - length of the chord be \(y=\sqrt{15^{2}-8^{2}}=\sqrt{225 - 64}=\sqrt{161}\approx12.7\). Then, using the geometric mean (or Pythagorean theorem in another way), we know that if a perpendicular from the center to a chord divides the chord into two segments. Let the two segments of the chord be \(a\) and \(b\). Here, \(a=x\), \(b\) can be found from the right - triangle with hypotenuse \(15\) and leg \(8\). Also, using the formula \(x=\sqrt{15^{2}-8^{2}}-8\) (another approach: using the property of right - angled triangles formed by radius, perpendicular to chord and half - chord).

$$x=\sqrt{15^{2}-8^{2}}-8$$

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Answer:

1. \(x\approx11.3\)
2. \(x\approx6.5\)
3. \(x\approx4.7\)