This problem involves finding the length of a chord in a larger circle that is tangent to a smaller concentric circle. We are given the radii of both concentric circles: the smaller circle has a radius of 12 cm, and the larger circle has a radius of 37 cm. To solve this, we will use fundamental geometric properties of circles and the Pythagorean theorem.

Concentric Circles Definition

Concentric circles are circles that share the same center point but have different radii. Imagine two or more circles drawn from the exact same central origin. In this problem, both the smaller and larger circles originate from the same central point.

A chord of a circle is a straight line segment whose endpoints both lie on the circle. In this specific scenario, the chord belongs to the larger circle.

A tangent to a circle is a straight line that touches the circle at exactly one point, without crossing into the interior of the circle. Here, the chord of the larger circle serves as a tangent to the smaller circle, meaning it touches the smaller circle at one specific point.

Key Geometric Principles for Circles

To accurately solve this problem, we rely on two important geometric principles:

• Radius to Tangent Property: When a radius is drawn from the center of a circle to the point where a tangent line touches the circle, this radius is always perpendicular to the tangent line. This creates a 90-degree angle, which is essential for forming a right-angled triangle.
• Chord Bisection Property: A perpendicular line segment drawn from the center of a circle to any chord within that circle will bisect (divide into two equal halves) the chord.

Pythagorean Theorem Application

Let's use the given information to set up our calculation:

• Let O be the common center of both concentric circles.
• Let the radius of the smaller circle be denoted as $r_1$. We are given $r_1 = 12 \text{ cm}$.
• Let the radius of the larger circle be denoted as $r_2$. We are given $r_2 = 37 \text{ cm}$.
• Let AB be the chord of the larger circle that is tangent to the smaller circle.
• Let M be the point where the chord AB touches (is tangent to) the smaller circle.

Based on the "Radius to Tangent Property," the radius OM (which is $r_1$) is perpendicular to the chord AB at point M. This forms a right-angled triangle, $\triangle OMA$, with the right angle at M.

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

• The side OM is the radius of the smaller circle, so $\text{OM} = 12 \text{ cm}$. This side represents one leg of the right triangle.
• The side OA is the radius of the larger circle, so $\text{OA} = 37 \text{ cm}$. This side is the hypotenuse (the longest side, opposite the right angle).
• The side AM is half the length of the chord AB. This side represents the other leg of the right triangle.

Now, we can apply the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): $a^2 + b^2 = c^2$.

For $\triangle OMA$, the theorem is:

$\text{OM}^2 + \text{AM}^2 = \text{OA}^2$

Substitute the known values into the equation:

$12^2 + \text{AM}^2 = 37^2$

Calculate the squares of the known values:

$144 + \text{AM}^2 = 1369$

To find $\text{AM}^2$, subtract 144 from both sides of the equation:

$\text{AM}^2 = 1369 - 144$

$\text{AM}^2 = 1225$

Finally, to find the length of AM, take the square root of 1225:

$\text{AM} = \sqrt{1225}$

$\text{AM} = 35 \text{ cm}$

Calculating Chord Length Precisely

As established by the "Chord Bisection Property," the perpendicular from the center O to the chord AB (which is OM) bisects the chord. This means that M is the midpoint of AB, and therefore AM is exactly half the total length of the chord AB.

To find the full length of the chord AB, we multiply the length of AM by 2:

Chord Length $\text{AB} = 2 \times \text{AM}$

Substitute the calculated value of AM:

Chord Length $\text{AB} = 2 \times 35 \text{ cm}$

Chord Length $\text{AB} = 70 \text{ cm}$

Step-by-Step Chord Length Calculation

Here’s a summary of the steps taken to determine the length of the chord:

1. Identify Radii: We identified the radius of the smaller circle ($r_1 = 12 \text{ cm}$) and the radius of the larger circle ($r_2 = 37 \text{ cm}$).
2. Form a Right Triangle: We recognized that the radius of the smaller circle (12 cm) drawn to the point of tangency (M) is perpendicular to the chord. Connecting the center (O) to an endpoint of the chord on the larger circle (A) forms the hypotenuse (37 cm) of a right-angled triangle $\triangle OMA$.
3. Apply Pythagorean Theorem: We used the Pythagorean theorem $(\text{OM}^2 + \text{AM}^2 = \text{OA}^2)$ to find the length of AM. This translated to $12^2 + \text{AM}^2 = 37^2$.
4. Solve for Half Chord Length (AM):
   • $144 + \text{AM}^2 = 1369$
   • $\text{AM}^2 = 1369 - 144$
   • $\text{AM}^2 = 1225$
   • $\text{AM} = \sqrt{1225} = 35 \text{ cm}$
5. Calculate Full Chord Length (AB): Since AM is half the chord length, we doubled it to find the total length of the chord. $\text{AB} = 2 \times 35 \text{ cm} = 70 \text{ cm}$.

Thus, the length of the chord of the larger circle that is tangent to the smaller circle is 70 cm.