Questions: The segments in each figure are tangent to the circle at the points shown. Find each length.

Transcript text: The segments in each figure are tangent to the circle at the points shown. Find each length.

Solution

Step 1: Set up the equation for problem 1

The tangent segments from a point outside a circle to the points of tangency are congruent. Therefore, $x+13 = 4x - 8$.

Step 2: Solve for x

Subtract x from both sides: $13 = 3x - 8$. Add 8 to both sides: $21 = 3x$. Divide by 3: $x=7$.

Step 3: Calculate AB

$AB = x+13 = 7+13 = 20$.

Step 4: Calculate BC

$BC = 4x-8 = 4(7) - 8 = 28-8=20$.

Step 5: Set up the equation for problem 2

Tangent segments from an external point to a circle are congruent. Therefore, $5x-11 = x+5$.

Step 6: Solve for x

Subtract x from both sides: $4x-11=5$. Add 11 to both sides: $4x = 16$. Divide both sides by 4: $x=4$.

Step 7: Calculate BD

$BD = 5x - 11 = 5(4)-11 = 20-11=9$.

Step 8: Calculate CD

$CD = x+5 = 4+5 = 9$.

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