Circles and Geometric Shapes | Exercise 5.3

Question 3

Two parallel chords of lengths $6\text{ cm}$ and $8\text{ cm}$ are on opposite sides of the centre of a circle. If the radius of the circle is $5\text{ cm}$ , find the distance between the midpoints of the chords.

Solution

Understand the Question

The perpendicular drawn from the centre of a circle to a chord bisects the chord.
Using the radius ( $r = 5\text{ cm}$ ) and half the length of each chord, we form right-angled triangles to find the perpendicular distance of each chord from the centre using the Pythagoras theorem.
Since the two parallel chords lie on opposite sides of the centre, the total distance between their midpoints is the sum of their individual distances from the centre: $\text{Distance} = d_1 + d_2$ .

Step 1 · Find Distance of First Chord from Centre

Let the chord be $AB = 6\text{ cm}$ and radius $r = 5\text{ cm}$ .

The perpendicular from the centre bisects the chord, so half the chord length is: $\dfrac{6}{2} = 3\text{ cm}$ Diagram 1

Let $d_1$ be the distance from the centre to chord $AB$ . By Pythagoras theorem:

\begin{aligned} d_1^2 + 3^2 &= 5^2 \\ d_1^2 + 9 &= 25 \\ d_1^2 &= 25 - 9 \\ d_1^2 &= 16 \\ d_1 &= 4\text{ cm} \end{aligned}

Step 2 · Find Distance of Second Chord from Centre

Let the chord be $CD = 8\text{ cm}$ and radius $r = 5\text{ cm}$ .

The perpendicular from the centre bisects the chord, so half the chord length is: $\dfrac{8}{2} = 4\text{ cm}$ Diagram 2

Let $d_2$ be the distance from the centre to chord $CD$ . By Pythagoras theorem:

\begin{aligned} d_2^2 + 4^2 &= 5^2 \\ d_2^2 + 16 &= 25 \\ d_2^2 &= 25 - 16 \\ d_2^2 &= 9 \\ d_2 &= 3\text{ cm} \end{aligned}

Step 3 · Calculate Total Distance between Midpoints

Since the chords are on opposite sides of the centre, the distance between their midpoints is the sum of $d_1$ and $d_2$ :

\begin{aligned} \text{Distance} &= d_1 + d_2 \\ &= 4 + 3 \\ &= 7\text{ cm} \end{aligned}

Answer

$7\text{ cm}$

Common Mistakes

Opposite vs. Same Side: Confusing opposite sides with same side. If the chords are on the same side, subtract the distances ( $4 - 3 = 1\text{ cm}$ ). Since they are on opposite sides, add them ( $4 + 3 = 7\text{ cm}$ ).
Using Full Chord Length: Forgetting to bisect the chord length before applying the Pythagoras theorem (using $6\text{ cm}$ and $8\text{ cm}$ directly instead of half-lengths $3\text{ cm}$ and $4\text{ cm}$ ).

Question 3

More questions in Exercise 5.3