1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Alex_Xolod [135]
2 years ago
13

Equilateral triangle ABC has an area of \sqrt{3}√ ​3 ​ ​​ . If the shaded region has an area of \piπK − \sqrt{3}√ ​3 ​ ​​ , what

is the value of K?
Mathematics
1 answer:
Liono4ka [1.6K]2 years ago
7 0

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)

∴ 4=r^{2}+r^{2}-2(r)(r)(-0.5)

∴ 4=r^{2}+r^{2}-(-r^{2})

∴ 4=r^{2}+r^{2}+r^{2}

∴ 4=3r^{2}

- Divide both sides by 3

∴ r^{2}=\frac{4}{3}

- 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

You might be interested in
PLEASE HELP LIKE PLEASEEEE
Yakvenalex [24]

Answer:

the answer is 5

plz mark me brainly

Step-by-step explanation:

3 0
3 years ago
24 is what percent of 49?
otez555 [7]
24 is what percent of 49? The answer to that is 48.98.
5 0
3 years ago
Read 2 more answers
The linear equation y = 1/4x tells us that this line is going to be more vertical or horizontal?
Afina-wow [57]
Horizontal, it’s below a 1/2 slope.
5 0
3 years ago
One meter is approximately equal to 39.39 inches. About how many inches are in 3 meters
mojhsa [17]
Well if 1 is 39.39 
39.39 times 3 is equal to 118.17 

don't forget to get my answer 5 satrs and thank me- Justin
8 0
2 years ago
Read 2 more answers
Calculate the distance between the points G= -2.2) and C = (5, -1) in the coordinate plane.
frozen [14]

Answer:

Step-by-step explanation:

4 0
1 year ago
Other questions:
  • If one endpoint of a segment is −3,−2 ( ) and the midpoint is −1,−8 ( ), find the other endpoint.
    13·1 answer
  • Two trains leave the same station at the same time, but are headed in opposite directions. One train travels at 70 mph and the o
    13·2 answers
  • What is the degree of the polynomial?
    11·1 answer
  • 3/8 times blank= 1 how do you solve this?
    11·1 answer
  • Scott was going to sell all of his stamp collection to buy a video game. After selling half of them he changed his mind. He boug
    14·1 answer
  • Which of the following is the solution to the inequality below -6(4-x)&lt;-4(x+1)
    11·1 answer
  • Harry buys a TV priced at £1200 plus 20% VAT.
    8·1 answer
  • Lakeisha spent $4.57 on color and black-and-white copies for her project. She made 7 more black-and-white copies than color copi
    10·1 answer
  • How do you solve this
    7·1 answer
  • Can somebody help please?????????
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!