Question 50 of 50advanced🔧 ApplyLong Answer5 marks

In a circle, two chords CC′CC' and DD′DD' are drawn perpendicular to a diameter ABAB. Prove that the segment MM′MM' joining the midpoints of the chords CDCD and C′D′C'D' is perpendicular to ABAB.

Correct Answer

Let ABAB be the diameter of the circle with centre OO. Since ABAB passes through OO and is perpendicular to the chords CC′CC' and DD′DD', by the converse of the given theorem, ABAB bisects CC′CC' and DD′DD'. Thus, the midpoints of CC′CC' and DD′DD' lie on ABAB.

We can consider ABAB as the x-axis with OO as the origin. Since CC′CC' and DD′DD' are perpendicular to ABAB, they are vertical lines. Let the x-coordinate of CC and C′C' be aa, and the x-coordinate of DD and D′D' be bb. The points lie on the circle, so their coordinates are C(a,y1)C(a, y_1), C′(a,−y1)C'(a, -y_1), D(b,y2)D(b, y_2), and D′(b,−y2)D'(b, -y_2).

Let MM be the midpoint of the chord CDCD. Using the midpoint formula, the x-coordinate of MM is:

a+b2\frac{a + b}{2}

Similarly, let M′M' be the midpoint of the chord C′D′C'D'. The x-coordinate of M′M' is also:

a+b2\frac{a + b}{2}

Because both midpoints MM and M′M' have the exact same x-coordinate, the line segment MM′MM' joining them is a vertical line. A vertical line is always perpendicular to the x-axis.

Since the diameter ABAB lies along the x-axis, the segment MM′MM' is perpendicular to ABAB. Hence proved.

Exercise: END-OF-CHAPTER EXERCISES | Q: *25 | (Page 25)
For More Understanding

Explanation

The proof combines the geometric property from the context (a perpendicular from the centre bisects the chord) with coordinate geometry. By placing the diameter ABAB on the x-axis, the perpendicular chords CC′CC' and DD′DD' become vertical lines, meaning their endpoints share x-coordinates.

Using the midpoint formula, the x-coordinates of the midpoints MM (of CDCD) and M′M' (of C′D′C'D') are identical, making MM′MM' a vertical line. Since a vertical line is perpendicular to the horizontal x-axis (ABAB), the statement is proven.

This approach is rigorous and aligns with Class 9 coordinate geometry methods.

Solution Steps

  1. Step 1: Let ABAB be the diameter along the x-axis with centre OO at the origin. By the converse of the given theorem, ABAB bisects CC′CC' and DD′DD'.

  2. Step 2: Since CC′CC' and DD′DD' are perpendicular to ABAB, they are vertical lines. Let C(a,y1)C(a, y_1), C′(a,−y1)C'(a, -y_1), D(b,y2)D(b, y_2), and D′(b,−y2)D'(b, -y_2).

  3. Step 3: Calculate the x-coordinate of midpoint MM of CDCD as a+b2\dfrac{a + b}{2}.

  4. Step 4: Calculate the x-coordinate of midpoint M′M' of C′D′C'D' as a+b2\dfrac{a + b}{2}.

  5. Step 5: Conclude that since MM and M′M' share the same x-coordinate, MM′MM' is a vertical line, making it perpendicular to the horizontal diameter ABAB.

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