Question : A circle is inscribed in the equilateral triangle \(\Delta\). What is the radius of the circle ?
Statement-I : Area of \(\Delta\) is equal to \(16\sqrt{3} \text{ cm}^2\).
Statement-II : Perimeter of \(\Delta\) is 24 cm.
Which one of the following is correct in respect of the above Question and the Statements ?
Analyzing Statement I: Equilateral Triangle Area
The question asks for the radius of a circle inscribed in an equilateral triangle (\(\Delta\)). Let the side length of the triangle be '\(a\)' and the inradius be '\(r\)'.
Statement I provides the area of \(\Delta\) as \(16\sqrt{3} \text{ cm}^2\). The formula for the area of an equilateral triangle is:
\(\text{Area} = \frac{\sqrt{3}}{4} a^2\)Using the given area:
\(16\sqrt{3} = \frac{\sqrt{3}}{4} a^2\)Solving for '\(a\)':
\(a^2 = \frac{4 \times 16\sqrt{3}}{\sqrt{3}} = 64\) \(a = \sqrt{64} = 8 \text{ cm}\)The inradius '\(r\)' of an equilateral triangle can be calculated using the side length '\(a\)' with the formula:
\(r = \frac{a}{2\sqrt{3}}\)Substituting \(a = 8\) cm:
\(r = \frac{8}{2\sqrt{3}} = \frac{4}{\sqrt{3}} = \frac{4\sqrt{3}}{3} \text{ cm}\)Conclusion: Statement I alone is sufficient to determine the side length and subsequently the inradius of the circle.
Analyzing Statement II: Equilateral Triangle Perimeter
Statement II provides the perimeter of \(\Delta\) as 24 cm.
The formula for the perimeter of an equilateral triangle is:
\(\text{Perimeter} = 3a\)Using the given perimeter:
\(24 = 3a\)Solving for '\(a\)':
\(a = \frac{24}{3} = 8 \text{ cm}\)Using the inradius formula for an equilateral triangle (\(r = \frac{a}{2\sqrt{3}}\)) with \(a = 8\) cm:
\(r = \frac{8}{2\sqrt{3}} = \frac{4}{\sqrt{3}} = \frac{4\sqrt{3}}{3} \text{ cm}\)Conclusion: Statement II alone is sufficient to determine the side length and subsequently the inradius of the circle.
Final Conclusion
Both Statement I and Statement II independently provide enough information to calculate the side length of the equilateral triangle, which allows for the calculation of the inscribed circle's radius. Therefore, the question can be answered using either statement alone.
Important Questions from Geometry
The sides of a triangle are in the ratio 6 : 4 : 3 and its perimeter is 104 cm. The length of the longest side (in cm) is:
An isosceles right-angled triangle has hypotenuse length as 10 units. What is the area of the triangle (in square units)?
Two circles of radii 16 cm and 4 cm, respectively, touch each other externally at Point A. PQ is the direct common tangent of these circles with centres C1 and C2, respectively. What is the length of PQ?
Let C be a circle with center O and AB be a chord of C such that the length of AB is equal to the radius of C. Let D be any point on the major arc of AB. Find ∠AOB and ∠ADB, respectively.
The centres of two circles are 84 cm apart. If the radii of these two circles are 38 cm and 26 cm, respectively, then which of the following options gives the length (in cm) of a direct common tangent of these two circles?
