Question 27 of 50beginner🔧 ApplyNumerical3 marks

In a circle, a chord is 5 cm away from the centre. If the radius of the circle is 13 cm, what is the length of the chord?

Correct Answer

24 cm

Exercise: END-OF-CHAPTER EXERCISES | Q: 1 | (Page 23)
For More Understanding

Explanation

According to Theorem 5 from the textbook, the perpendicular from the centre of a circle to a chord bisects the chord. This creates a right-angled triangle where the radius is the hypotenuse, the perpendicular distance from the centre to the chord is one leg, and half the length of the chord is the other leg. Using the Pythagorean theorem, the chord length formula is derived as 2r2−d2\sqrt{r² - d²}. Substituting the given values of radius (13 cm) and distance (5 cm) into this formula yields the length of the chord.

Solution Steps

  1. Step 1: Identify the given values. Radius (r) = 13 cm, Perpendicular distance from centre (d) = 5 cm.

  2. Step 2: Apply the formula for the length of the chord. Length of chord = 2r2−d2\sqrt{r² - d²}.

  3. Step 3: Substitute the values into the formula. Length of chord = 2132−52\sqrt{13² - 5²}.

  4. Step 4: Calculate the squares. 13² = 169 and 5² = 25.

  5. Step 5: Subtract the values under the square root. 169 - 25 = 144.

  6. Step 6: Find the square root. 144\sqrt{144} = 12. This is half the length of the chord.

  7. Step 7: Multiply by 2 to find the full length of the chord. 2×122 \times 12 = 24 cm.

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