“There is no chord of a circle that is longer than its diameter.” How do you justify this statement?
We know that the longer the chord, the closer it is to the centre of the circle. This implies that as a chord's length increases, its perpendicular distance from the centre decreases.
The chord that is nearest to the centre is the one that contains the centre itself. For this specific chord, the perpendicular distance from the centre is zero.
The chord passing through the centre of the circle is the diameter. Since the diameter is the chord closest to the centre, it must be the greatest chord in the circle. Hence, there is no chord of a circle that is longer than its diameter.
Explanation
The textbook establishes the relationship between chord length and its distance from the centre using Theorem 8, which states that longer chords are closer to the centre. The comment following the theorem highlights that the chord nearest to the centre contains the centre (distance = 0), which is the diameter. Thus, the diameter is the greatest chord.
Solution Steps
Step 1: State the property that the longer the chord, the closer it is to the centre.
Step 2: Identify that the chord nearest to the centre passes through the centre (distance = 0), which is the diameter.
Step 3: Conclude that since the diameter is closest to the centre, it is the greatest chord, so no chord is longer than it.