Question

Two concentric circles are of radii $5\text{ cm}$ and $4\text{ cm}$. Find the length of the chord of the larger circle which touches the smaller circle.

Hint:

A perpendicular line from the center of concentric circles to a chord of the outer circle that is tangent to the inner circle always bisects the chord.
The chord length is always:
\[ L = 2\sqrt{R^2 - r^2} \]
where $R$ is the outer radius and $r$ is the inner radius.
Using this directly:
\[ L = 2\sqrt{5^2 - 4^2} = 2\sqrt{9} = 6\text{ cm} \]

Answer 1

Step 1: Solve this with coordinates instead of the classical circle theorem approach.
Rather than drawing triangle $OPA$ and applying Pythagoras' theorem inside it, let's place the whole figure on a coordinate plane and find the chord as the intersection of a line with a circle.

Step 2: Set up coordinates.
Place the common centre $O$ of the two concentric circles at the origin $(0, 0)$. The smaller circle then has the equation $x^2 + y^2 = 4^2 = 16$, and the larger circle has the equation $x^2 + y^2 = 5^2 = 25$.

Step 3: Place the tangent chord using its point of contact.
Since the chord of the larger circle touches the smaller circle, it touches the smaller circle at exactly one point, and the radius to that point is perpendicular to the chord. Without loss of generality, take the point of contact to be $(4, 0)$ on the smaller circle (this is always possible by rotating our coordinate axes to line up with that point). Since the radius $O(4,0)$ lies along the x-axis, the tangent chord at this point must be a vertical line, so its equation is:
\[ x = 4 \]

Step 4: Find where this vertical line cuts the larger circle.
Substitute $x = 4$ into the equation of the larger circle:
\[ 4^2 + y^2 = 25 \]
\[ 16 + y^2 = 25 \]
\[ y^2 = 9 \implies y = \pm 3 \]
So the chord meets the larger circle at the two points $(4, 3)$ and $(4, -3)$.

Step 5: Compute the length of the chord.
Since both points share the same $x$-coordinate, the chord is vertical, and its length is simply the difference in $y$-coordinates:
\[ \text{Length} = 3 - (-3) = 6\text{ cm} \]

Step 6: Final answer.
The length of the chord of the larger circle is $6\text{ cm}$.
\[ \boxed{6\text{ cm}} \]