# How to find the angles of a triangle in a semicircle?

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1. Jun 14, 2017

### Helly123

1. The problem statement, all variables and given/known data

2. Relevant equations
d(y)/d(x) --> max area
area of triangle = 1/2 . base . height

3. The attempt at a solution
for number (2)

x^2 + y^2 = r^2 --> circle equation
base = 2R, height = y
Area = 1/2 . 2R . y
area = 1/2 . 4. √ (r^2 - x^2)

area now is half of max = 2, so :
2 = 1/2 . 4. √ (r^2 - x^2)
2 = 2 √(4-x^2)
1 = √ (4-x^2)
1 = 4 - x^2
x = √3
now y = 1
AP = 2 - √3
I have to find w (angle of ACB)
PC = 1
how to find w ?

2. Jun 14, 2017

### Staff: Mentor

Consider the triangle AOC.

3. Jun 15, 2017

### Helly123

OC^2 = AC^2 + AO^2 -2AC.AO.cosw
4 = 4 + (8-4v3)^2 -8(8-4v3)cosw
Cos w = (8-4v3)/8
Cos w = 1-1/2v3

But the answer is in degrees, and not allowed using calculator.. And I dont think my answer is right.

Note: v3 = root 3

4. Jun 15, 2017

### Staff: Mentor

Consider the triangle AOC. You can find the angle at O by studying the right triangle CPO.

5. Jun 15, 2017

### Helly123

I don't get it.. Can anyone help??

6. Jun 15, 2017

### ehild

CPO is a right triangle. You know the hypotenuse, it is one radius, length 2. You calculated PC, it is 1. What is the blue angle then?

7. Jun 15, 2017

### Helly123

The blue angle is 60degrees.

8. Jun 15, 2017

### ehild

No. Imagine half of an equilateral triangle, rotated

9. Jun 15, 2017

### Helly123

Oh.. 30degrees ...

10. Jun 15, 2017

### Helly123

So, i get it. One triangle = 180degrees. I get 30, left 150. CAO = ACO. So CAO = 75 degrees?

11. Jun 15, 2017

### Helly123

But with theory, it should be sin w/ 2 = sin tetha / AC
I get sin w = 1/AC
While AC = 8-4root3
How can i get w with this method?

12. Jun 16, 2017

### ehild

Yes, it is correct.

13. Jun 16, 2017

### ehild

What is theta, and what "theory" is that?
Check with a calculator. Otherwise it is complicated.