Answer:
The value of k = 4/3
Step-by-step explanation:
* Lets explain how to solve the problem
- An equilateral triangle ABC is inscribed in a circle N
- The area of the triangle is √3
- The shaded area is the difference between the area of the circle
and the area of the equilateral triangle ABC
- The shaded are = k π - √3
- We need to find the value of k
* <u><em>At first lets find the length of the side of the Δ ABC</em></u>
∵ Δ ABC is an equilateral triangle
∴ Its area = √3/4 s² , where s is the length of its sides
∵ The area of the triangle = √3
∴ √3/4 s² = √3
- divide both sides by √3
∴ 1/4 s² = 1
- Multiply both sides by 4
∴ s² = 4 ⇒ take √ for both sides
∴ s = 2
∴ The length of the side of the equilateral triangle is 2
* <u><em>Now lets find the radius of the circle</em></u>
- In the triangle whose vertices are A , B and N the center of the circle
∵ AN and BN are radii
∴ AN = BN = r , where r is the radius of the circle
∵ The sides of the equilateral angles divides the circle into 3 equal
arcs in measure where each arc has measure 360°/3 = 120°
∵ The measure of the central angle in a circle equal the measure
of the its subtended arc arc
∵ ∠ANB is an central angle subtended by arc AB
∵ The measure of arc AB is 120°
∴ m∠ANB = 120°
- By using the cosine rule in Δ ANB
∵ AB = 2 , AN = BN = r , m∠ANB = 120°
∴ ![(2)^{2}=r^{2}+r^{2}-2(r)(r)cos(120)](https://tex.z-dn.net/?f=%282%29%5E%7B2%7D%3Dr%5E%7B2%7D%2Br%5E%7B2%7D-2%28r%29%28r%29cos%28120%29)
∴ ![4=r^{2}+r^{2}-2(r)(r)(-0.5)](https://tex.z-dn.net/?f=4%3Dr%5E%7B2%7D%2Br%5E%7B2%7D-2%28r%29%28r%29%28-0.5%29)
∴ ![4=r^{2}+r^{2}-(-r^{2})](https://tex.z-dn.net/?f=4%3Dr%5E%7B2%7D%2Br%5E%7B2%7D-%28-r%5E%7B2%7D%29)
∴ ![4=r^{2}+r^{2}+r^{2}](https://tex.z-dn.net/?f=4%3Dr%5E%7B2%7D%2Br%5E%7B2%7D%2Br%5E%7B2%7D)
∴ ![4=3r^{2}](https://tex.z-dn.net/?f=4%3D3r%5E%7B2%7D)
- Divide both sides by 3
∴ ![r^{2}=\frac{4}{3}](https://tex.z-dn.net/?f=r%5E%7B2%7D%3D%5Cfrac%7B4%7D%7B3%7D)
- Take square root for both sides
∴ r = 2/√3
* <u><em>Lets find the value of k</em></u>
∵ Area circle = πr²
∵ r = 2/√3
∴ Area circle = π(2/√3)² = (4/3)π
∵ Area shaded = area circle - area triangle
∵ Area triangle = √3
∴ Area shaded = (4/3) π - √3
∵ Area of the shaded part is π k - √3
- Equate the two expressions
∴ π k - √3 = (4/3) π - √3
∴ k = 4/3
* The value of k = 4/3