AB is a chord of length 24cm of a circle of radius 13cm. The tangents at A and B intersect at a point C. Find the length AC.

To find the length of AC, we can use the properties of tangents and chords in a circle. Here’s how to approach the problem:

Step 1: Identify Key Elements

We have a circle with:

• Radius (r) = 13 cm
• Chord AB = 24 cm

Step 2: Calculate the Distance from the Center to the Chord

Let O be the center of the circle. The perpendicular distance from the center O to the chord AB can be found using the formula:

d = √(r² - (c/2)²)

Where:

• d = distance from the center to the chord
• r = radius of the circle
• c = length of the chord

Substituting the values:

d = √(13² - (24/2)²) = √(169 - 144) = √25 = 5 cm

Step 3: Use the Tangent-Secant Theorem

The Tangent-Secant Theorem states that the square of the length of a tangent segment (AC) from an external point to a circle is equal to the product of the lengths of the entire secant segment (AB) and its external part (AC).

Let AC = x. Then, according to the theorem:

x² = AB × (AB + 2d)

Here, AB = 24 cm and d = 5 cm:

x² = 24 × (24 + 2 × 5) = 24 × (24 + 10) = 24 × 34 = 816

Step 4: Solve for AC

Now, take the square root to find AC:

AC = √816 ≈ 28.6 cm

Final Answer

The length of AC is approximately 28.6 cm.