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A circle with two chords is shown below. What is the value of x? Round your answer to the nearest tenth. Do not include "x=" in your answer.
numbers given are 8 20 and 36
Explain Bot
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To find the value of x in the circle with two chords, we can make use of the properties of chords in a circle.
Step 1: Let's label the points where the chords intersect the circle as A, B, C, and D, as shown in the diagram.
Step 2: Recall that when two chords intersect inside a circle, the product of the segments of one chord is equal to the product of the segments of the other chord. In equation form, this can be written as:
AC * BD = AD * BC
Step 3: Apply the property to the given information:
AC * BD = AD * BC
We are given the lengths of the chords: AB = 8, CD = 20, and BC = 36. Let's assume that x is the unknown segment AD.
Now, we have two segments of one chord: AD and BD. The other two segments of the other chord are AC and BC.
To solve for x, we need to set up the equation using the given information:
x * (x + 20) = 8 * 36
Step 4: Simplify the equation and solve for x:
x^2 + 20x = 288
Rearranging the equation to standard quadratic form:
x^2 + 20x - 288 = 0
Step 5: Solve the quadratic equation using factoring, completing the square, or the quadratic formula. In this case, the quadratic factors as:
(x + 24)(x - 4) = 0
Setting each factor equal to zero:
x + 24 = 0 or x - 4 = 0
Solving for x, we have:
x = -24 or x = 4
Since we are looking for a positive value for x, we can disregard the negative value.
Therefore, the value of x is 4 rounded to the nearest tenth. Hence, x = 4.