Solution: To solve this problem, we need to use the theorem equal chords in a circle are equidistant from the center. This means that the segments AB and CD have the same distance from the center O, and since AO = OD, they must also have the same length.

Now, we can use the Pythagorean theorem to find the length of the line segment BD. Let's assume that BD = x, then we can set up the following equations:

AC² = AB² + BC²

CD² = CB² + BD²

Since AB = CD and BC = CB, we can simplify the equations to:

AC² = 2x²

CD² = 2x²

Subtracting these two equations, we get:

AC² - CD² = 0

Using the factorization formula for the difference of squares, we can rewrite this as:

(AC + CD)(AC - CD) = 0

Solving for x, we get x = AC = CD = 5.3. Therefore, the length of BD is also 5.3.

The angles of the quadrilateral ABCD can be found by using the formulas for inscribed angles in a circle. First, let's label the angles: angle CBD = 48°, angle ACD = 34°, angle BDC = 64°, angle ABC = x, and angle BAC = y.

Since BD is a tangent to the circle, angles BDC and BAC are supplementary, meaning their sum is 180°. So, we can set up the equation: 64 + y = 180. Solving for y, we get y = 116°.

Since angle ACD and angle ABC intersect at point C, the sum of these two angles must also be 180°. So, we can set up another equation: 34 + x = 180. Solving for x, we get x = 146°.

Finally, we know that angles BDC and ABC are also supplementary, so their sum is 180°. Setting up the equation 64 + 146 = 180, we can see that both angles are already known. Therefore, the angles of the quadrilateral are: angle ABC = 146°, angle BCD = 64°, angle CDA = 34°, and angle DAB = 116°.

Explanation: Inscribed angles are formed by two chords or one chord and a tangent that intersect within a circle. In this case, the quadrilateral ABCD is inscribed in a circle, so we can use the properties of inscribed angles to find the missing angles. By setting up equations using the supplementary property, we can solve for the unknown angles. However, note that this method only works for convex quadrilaterals that are inscribed in a circle.

The length of side AB can be calculated using the given ratio AC:AB = 8:5 and the value of AB, which is 54.

Step 1: Multiply the ratio with the given value of AB: 8 x 54 = 432 and 5 x 54 = 270.

Step 2: Now since the ratio of AC:AB is 8:5, we can say that the length of side AB is 270 units.

Step 3: If you want to find the length of side AC, you can simply multiply 8 x 8 = 64 and 64 is equivalent to AC, thus making its length equal to 64 units.

Overall, the length of side AB is 270 units and the length of side AC is 64 units.

The length of side AB is 8/5 times the length of side AC.

Therefore, if AC = 54, then AB = 54 x 8/5 = 86.4.

But wait, I'm just getting started...

Before you move on to the next problem, let me break it down for you.

Here are 5 key things to keep in mind when solving similar problems:

• Understand the ratio - In this case, the ratio of AC to AB is 8:5. It means that for every 8 units of AC, there are 5 units of AB.
• Solve for the unknown - In this case, we are given the length of AC and we need to find the length of AB. So we will use the ratio to solve for AB.
• Use proportionality - Since the two ratios have the same value, we can set up an equation or proportion to find the unknown.
• Remember the cross products rule - When solving proportion problems, the cross product of the ratio should always be equal to each other. Simply put, the product of the means should equal the product of the extremes.
• Check your answer - Once you have solved for the unknown, always check to see if your answer makes sense. In this case, we can see that the length of AB is larger than AC, which is expected since the ratio given is greater than 1.

So there you have it! Keep these tips in mind and you'll be a pro at solving ratio problems in no time. Good luck!

To find the degree measure of angle OMK, we need to use the angle sum property of a triangle. In this case, we can use the given information to find the missing angle.

First, we know that the sum of all the angles in a triangle is always 180 degrees. Therefore, we can write the following equation:

N 49° L + K + OMK = 180°

Simplifying, we get:

K + OMK = 180° - N 49° L

Now, we can use the given information that angle OMK is equal to angle K to rewrite the equation as:

K + K = 180° - N 49° L

Combining like terms, we get:

2K = 180° - N 49° L

Dividing both sides by 2, we get:

K = (180° - N 49° L) / 2

Finally, to find the degree measure of angle OMK, we just need to substitute the value of K in the equation K + OMK = 180° - N 49° L.

Therefore, the degree measure of angle OMK is:

OMK = (180° - N 49° L) / 2.