Recall that the perpendicular distance from the center of the circle to the chord will bisect the chord itself.

Thus, we can use the Pythagorean Theorem to find the length of the chord.

\(\displaystyle \text{Radius}^2=\text{distance}^2+\text{half chord}^2\)

Rearrange the equation to solve for the length of half the chord.

\(\displaystyle \text{Half Chord}=\sqrt{\text{Radius}^2-\text{Distance}^2}\)

Now, multiply this value by two to find the length of the entire chord.

\(\displaystyle \text{Chord}=2\sqrt{\text{Radius}^2-\text{Distance}^2}\)

Plug in the given radius and distance to find the length of the chord.

\(\displaystyle \text{Chord}=2\sqrt{10^2-6^2}=2\sqrt{64}=16\)

Recall that the perpendicular distance from the center of the circle to the chord will bisect the chord itself.

Thus, we can use the Pythagorean Theorem to find the length of the chord.

\(\displaystyle \text{Radius}^2=\text{distance}^2+\text{half chord}^2\)

Rearrange the equation to solve for the length of half the chord.

\(\displaystyle \text{Half Chord}=\sqrt{\text{Radius}^2-\text{Distance}^2}\)

Now, multiply this value by two to find the length of the entire chord.

\(\displaystyle \text{Chord}=2\sqrt{\text{Radius}^2-\text{Distance}^2}\)

Plug in the given radius and distance to find the length of the chord.

\(\displaystyle \text{Chord}=2\sqrt{9^2-4^2}=2\sqrt{65}=16.12\)

Make sure to round to two places after the decimal.

Recall that the perpendicular distance from the center of the circle to the chord will bisect the chord itself.

Thus, we can use the Pythagorean Theorem to find the length of the chord.

\(\displaystyle \text{Radius}^2=\text{distance}^2+\text{half chord}^2\)

Rearrange the equation to solve for the length of half the chord.

\(\displaystyle \text{Half Chord}=\sqrt{\text{Radius}^2-\text{Distance}^2}\)

Now, multiply this value by two to find the length of the entire chord.

\(\displaystyle \text{Chord}=2\sqrt{\text{Radius}^2-\text{Distance}^2}\)

Plug in the given radius and distance to find the length of the chord.

\(\displaystyle \text{Chord}=2\sqrt{24^2-19^2}=2\sqrt{215}=29.33\)

Make sure to round to two places after the decimal.

Recall that the perpendicular distance from the center of the circle to the chord will bisect the chord itself.

Thus, we can use the Pythagorean Theorem to find the length of the chord.

\(\displaystyle \text{Radius}^2=\text{distance}^2+\text{half chord}^2\)

Rearrange the equation to solve for the length of half the chord.

\(\displaystyle \text{Half Chord}=\sqrt{\text{Radius}^2-\text{Distance}^2}\)

Now, multiply this value by two to find the length of the entire chord.

\(\displaystyle \text{Chord}=2\sqrt{\text{Radius}^2-\text{Distance}^2}\)

Plug in the given radius and distance to find the length of the chord.

\(\displaystyle \text{Chord}=2\sqrt{5^2-1^2}=2\sqrt{24}=9.80\)

Make sure to round to two places after the decimal.

Recall that the perpendicular distance from the center of the circle to the chord will bisect the chord itself.

Thus, we can use the Pythagorean Theorem to find the length of the chord.

\(\displaystyle \text{Radius}^2=\text{distance}^2+\text{half chord}^2\)

Rearrange the equation to solve for the length of half the chord.

\(\displaystyle \text{Half Chord}=\sqrt{\text{Radius}^2-\text{Distance}^2}\)

Now, multiply this value by two to find the length of the entire chord.

\(\displaystyle \text{Chord}=2\sqrt{\text{Radius}^2-\text{Distance}^2}\)

Plug in the given radius and distance to find the length of the chord.

\(\displaystyle \text{Chord}=2\sqrt{27^2-20^2}=2\sqrt{329}=36.28\)

Make sure to round to two places after the decimal.

Recall that the perpendicular distance from the center of the circle to the chord will bisect the chord itself.

Thus, we can use the Pythagorean Theorem to find the length of the chord.

\(\displaystyle \text{Radius}^2=\text{distance}^2+\text{half chord}^2\)

Rearrange the equation to solve for the length of half the chord.

\(\displaystyle \text{Half Chord}=\sqrt{\text{Radius}^2-\text{Distance}^2}\)

Now, multiply this value by two to find the length of the entire chord.

\(\displaystyle \text{Chord}=2\sqrt{\text{Radius}^2-\text{Distance}^2}\)

Plug in the given radius and distance to find the length of the chord.

\(\displaystyle \text{Chord}=2\sqrt{13^2-5^2}=2\sqrt{144}=24\)

Recall that the perpendicular distance from the center of the circle to the chord will bisect the chord itself.

Thus, we can use the Pythagorean Theorem to find the length of the chord.

\(\displaystyle \text{Radius}^2=\text{distance}^2+\text{half chord}^2\)

Rearrange the equation to solve for the length of half the chord.

\(\displaystyle \text{Half Chord}=\sqrt{\text{Radius}^2-\text{Distance}^2}\)

Now, multiply this value by two to find the length of the entire chord.

\(\displaystyle \text{Chord}=2\sqrt{\text{Radius}^2-\text{Distance}^2}\)

Plug in the given radius and distance to find the length of the chord.

\(\displaystyle \text{Chord}=2\sqrt{12^2-4^2}=2\sqrt{128}=22.63\)

Make sure to round to two places after the decimal.

If two chords intersect inside a circle, they cut each other into segments such that the product of the lengths of the portions of one chord is equal to that of the lengths of the portions of the other chord. In other words, 

\(\displaystyle AX \cdot XB = CX \cdot XD\)

Set \(\displaystyle XB= 18, CX = 12, XD= 9\), and solve for \(\displaystyle AX\):

\(\displaystyle AX \cdot 18 = 12\cdot 9\)

\(\displaystyle AX \cdot 18 = 108\)

\(\displaystyle AX \cdot 18\div 18 = 108 \div 18\)

\(\displaystyle AX = 6\).

When a chord is intercepted by a perpendicular line segment originating at the center of the circle, the chord is bisected, or cut in half. From the figure, you should notice that the base of the triangle is also the chord of the circle.

Thus, you can use the Pythagorean theorem to find the half of the chord that acts as the other leg of the right triangle that is created.

\(\displaystyle \text{Half Chord}=\sqrt{\text{radius}^2-\text{leg}^2}\)

\(\displaystyle \text{Half Chord}=\sqrt{10^2-6^2}=\sqrt{64}=8\)

Multiply this by \(\displaystyle 2\) to find the length of the entire chord.

\(\displaystyle \text{Length of chord}=\text{Base of Triangle}=16\)

Next, find the area of the triangle.

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}bh\)

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}(16)(6)=48\)

Next, find the area of the circle.

\(\displaystyle \text{Area of Circle}=\pi r^{2}\)

\(\displaystyle \text{Area of Circle}=\pi\times 10^2=100\pi\)

Finally, find the area of the shaded region.

\(\displaystyle \text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}\)

\(\displaystyle \text{Area of Shaded Region}=100\pi-48=266.2 \text{ cm}^{2}\)

Two figures are congruent by definition if all of their corresponding sides are congruent and all of their corresponding angles are congruent.

By definition, a rhombus has four sides of equal length.  If we let \(\displaystyle s_{1}\) be the common sidelength of Rhombus \(\displaystyle ABCD\) and \(\displaystyle s_{2}\) be the common sidelength of Rhombus \(\displaystyle EFGH\), then, since \(\displaystyle AB = EF\), it follows that \(\displaystyle s_{1} = s_{2}\), so corresponding sides are congruent. However, no information is given about their angle measures. Therefore, it cannot be determined whether or not the two rhombuses are congruent.