Question

find the length of segment b.
\\(a = 5\text{ in}\\)
\\(c = 4\text{ in}\\)
\\(d = 20\text{ in}\\)
\\(b = ?\text{ in}\\)

Explanation:

Identify the geometric theorem

The diagram shows two secant segments drawn to a circle from an external point.
We apply the Intersecting Secants Theorem (also known as the Secant-Secant Power Theorem).

State the formula

For two secants from an external point, the product of the external segment and the entire secant segment is equal for both secants:

$$ \text{external}_1 \times \text{total}_1 = \text{external}_2 \times \text{total}_2 $$

Define the segment lengths

From the given image, we identify:

• First secant:
• External segment: \(a = 5\text{ in}\)
• Internal segment: \(b\)
• Total segment: \(a + b = 5 + b\)
• Second secant:
• External segment: \(c = 4\text{ in}\)
• Internal segment: \(d = 20\text{ in}\)
• Total segment: \(c + d = 4 + 20 = 24\text{ in}\)

Set up the equation

Substitute the values into the theorem's formula:

$$ a \cdot (a + b) = c \cdot (c + d) $$

$$ 5 \cdot (5 + b) = 4 \cdot (4 + 20) $$

Solve for b

Simplify and solve the algebraic equation:

$$ 5(5 + b) = 4(24) $$

$$ 25 + 5b = 96 $$

$$ 5b = 71 $$

$$ b = 14.2 $$

Answer:

Find the length of segment b.

\(b =\) <blank>14.2</blank> in