Identify the right-angled triangle and known lengths.

In the given diagram, O is the center of the circle, and AB is a chord. M is the midpoint of AB. The line segment OM connects the center to the midpoint of the chord. A property of circles states that the line from the center to the midpoint of a chord is perpendicular to the chord. Therefore, $\triangle OMA$ is a right-angled triangle with the right angle at M.
We are given:

• Radius $OA = 13$ units (hypotenuse of $\triangle OMA$)
• Distance from center to chord $OM = 5$ units (one leg of $\triangle OMA$) We need to find the length of $AM$ (the other leg of $\triangle OMA$).

For the right-angled triangle $\triangle OMA$:

$$OA^2 = OM^2 + AM^2$$

Substitute the known values:

$$13^2 = 5^2 + AM^2$$

$$169 = 25 + AM^2$$ $$AM^2 = 169 - 25$$ $$AM^2 = 144$$ $$AM = \sqrt{144}$$ $$AM = 12 \text{ units}$$

Since M is the midpoint of chord AB, the length of AB is twice the length of AM.

$$AB = 2 \times AM$$ $$AB = 2 \times 12$$ $$AB = 24 \text{ units}$$

Reasons:

• The line from the center of a circle to the midpoint of a chord is perpendicular to the chord. Thus, $\triangle OMA$ is a right-angled triangle at M.
• The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (radius OA) is equal to the sum of the squares of the other two sides (OM and AM).
• M is the midpoint of AB, so $AB = 2 \times AM$.

The length of chord AB is $\text{24 units}$.