Finding the Diameter of a Circle Using Chord Length and Perpendicular Distance

The problem asks us to find the diameter of a circle given the length of a chord and the perpendicular distance from the center of the circle to that chord. This is a classic geometry problem involving the properties of circles.

Understanding the Problem Setup

We are given:

• Length of the perpendicular from the center to the chord = 15 cm
• Length of the chord = 40 cm

We need to find the diameter of the circle.

Key Geometric Principle: Perpendicular from Center to Chord

A fundamental theorem in circle geometry states that a perpendicular drawn from the center of a circle to a chord bisects the chord. This means it divides the chord into two equal parts.

Calculating the Half-Chord Length

Given the chord length is 40 cm, the perpendicular from the center bisects it into two segments of equal length.

Half-chord length $$ = \frac{\text{Chord Length}}{2} $$

Half-chord length $$ = \frac{40 \text{ cm}}{2} $$

Half-chord length $$ = 20 \text{ cm} $$

Forming a Right-Angled Triangle

Consider the radius of the circle drawn to one end of the chord. This radius, the perpendicular from the center to the chord, and the half-chord form a right-angled triangle. The right angle is at the point where the perpendicular meets the chord.

• One leg of the right triangle is the perpendicular distance from the center to the chord (15 cm).
• The other leg is the half-chord length (20 cm).
• The hypotenuse of the right triangle is the radius of the circle (let's call it $$r$$).

Applying the Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The Pythagorean theorem is stated as $$a^2 + b^2 = c^2$$, where $$a$$ and $$b$$ are the lengths of the legs, and $$c$$ is the length of the hypotenuse.

In our case:

• Leg 1 $$= 15 \text{ cm}$$ (perpendicular distance)
• Leg 2 $$= 20 \text{ cm}$$ (half-chord length)
• Hypotenuse $$= r$$ (radius)

Using the Pythagorean theorem:

$$ r^2 = 15^2 + 20^2 $$

$$ r^2 = 225 + 400 $$

$$ r^2 = 625 $$

To find the radius $$r$$, we take the square root of 625:

$$ r = \sqrt{625} $$

$$ r = 25 \text{ cm} $$

Calculating the Diameter

The diameter of a circle is twice its radius.

Diameter $$ = 2 \times \text{Radius} $$

Diameter $$ = 2 \times 25 \text{ cm} $$

Diameter $$ = 50 \text{ cm} $$

Final Answer

The diameter of the circle is 50 cm.

Revision Table: Circle Geometry Concepts

Additional Information on Circle Properties and Calculations

Problems involving chords, radii, and the center of a circle often require understanding the relationships between these elements. The property that a perpendicular from the center bisects the chord is very useful.

• If you know the radius and the distance of a chord from the center, you can find the chord length using the Pythagorean theorem.
• If you know the radius and the chord length, you can find the distance of the chord from the center using the Pythagorean theorem.
• The longest chord in a circle is the diameter.

Mastering these concepts helps in solving various circle geometry problems.