A chord of length 5 cm subtends an angle of \(90^\circ\) at the centre. Find the difference between the areas of the two segments formed by the chord.
\(\dfrac{25}{4}(\pi+2)\,\text{cm}^2 \;\approx\; 32.14\,\text{cm}^2\)
Step 1: Understand the problem.
A circle has a chord of length 5 cm. This chord makes an angle of \(90^\circ\) at the centre. The chord divides the circle into two parts called segments (minor and major). We need to find the difference between the areas of these two segments.
Step 2: Relation between chord length and radius.
Formula: For a chord of length \(c\), subtending angle \(\theta\) at the centre: \[ c = 2r \sin\left(\dfrac{\theta}{2}\right) \]
Here, \(c = 5\,\text{cm}\), \(\theta = 90^\circ\). So, \(c = 2r \sin 45^\circ = 2r \cdot \dfrac{1}{\sqrt{2}} = r\sqrt{2}\). Therefore, \[ r = \dfrac{5}{\sqrt{2}}\,\text{cm} \]
Step 3: Area of the sector (angle = 90°).
Formula: \(\text{Sector area} = \dfrac{\theta}{360^\circ} \times \pi r^2\). Substituting \(\theta = 90^\circ\): \[ \text{Sector area} = \dfrac{90}{360} \pi r^2 = \dfrac{1}{4}\pi r^2 \]
With \(r^2 = (\tfrac{5}{\sqrt{2}})^2 = \tfrac{25}{2}\): \[ \text{Sector area} = \dfrac{1}{4}\pi \times \dfrac{25}{2} = \dfrac{25\pi}{8}\,\text{cm}^2 \]
Step 4: Area of the triangle (isosceles, two sides = radius).
Formula: \(\text{Triangle area} = \tfrac{1}{2}r^2 \sin \theta\). Here, \(\theta = 90^\circ\), so \(\sin 90^\circ = 1\). \[ \text{Triangle area} = \tfrac{1}{2} \times \tfrac{25}{2} = \dfrac{25}{4}\,\text{cm}^2 \]
Step 5: Area of the minor segment.
Minor segment = Sector area – Triangle area \[ = \dfrac{25\pi}{8} - \dfrac{25}{4} = \dfrac{25}{8}(\pi - 2)\,\text{cm}^2 \]
Step 6: Area of the major segment.
Total circle area = \(\pi r^2 = \pi \times \tfrac{25}{2} = \dfrac{25\pi}{2}\,\text{cm}^2\). Major segment = Total circle area – Minor segment \[ = \dfrac{25\pi}{2} - \dfrac{25}{8}(\pi - 2) = \dfrac{25}{8}(3\pi + 2)\,\text{cm}^2 \]
Step 7: Difference of areas.
Difference = Major segment – Minor segment \[ = \dfrac{25}{8} \big[(3\pi + 2) - (\pi - 2)\big] \] Simplify inside the bracket: \(3\pi + 2 - \pi + 2 = 2\pi + 4\). So, \[ \text{Difference} = \dfrac{25}{8} \times (2\pi + 4) = \dfrac{25}{4}(\pi + 2)\,\text{cm}^2 \]
Step 8: Approximate value.
Taking \(\pi \approx 3.14\): \[ \text{Difference} = \dfrac{25}{4}(3.14 + 2) = \dfrac{25}{4} \times 5.14 \approx 32.14\,\text{cm}^2 \]
Final Answer: Difference between the areas of the two segments = \(\dfrac{25}{4}(\pi+2)\,\text{cm}^2 \;\approx\; 32.14\,\text{cm}^2\).