Find the length of the chord PQ .

Question

P and Q are both points on the unit circle. The standard angle of P is 45 degrees and the standard angle of Q is 150 degrees. Find the length of the chord PQ. Round answer to 2 decimal places.

Answer 1

 

On the unit circle ...

 

At 45º, the coordinates of the point P are (√2/2, √2/2)

 

At 150º, the coordinates of point Q are (-√3/2, 1/2)

 

We can use the distance formula to find the length of PQ

 

   Let P be point 1, and Q be point 2

 

Distance = √((x₂ - x₁)² + (y₂ - y₁)²)

             = √((-√3/2 - √2/2)² + (1/2 - √2/2)²)

             = √ ((-1.57313)² + (-.20711)²)

             =√(2.4747 + .04289)

             =√2.5176

             = 1.59

 

Thius the length of PQ is 1.59

 

Answer 2

We can solve this using the law of sines.  The angle (150-45) = 105 degrees forms a triangle with 2 equal angles that add up to (180-105) = 75 degrees.  Hence, the 2 angles are 37.5 degrees.  We then have a triangle with a side of length x opposite 105 degrees, and equal sides of length 1 (unit circle) opposite 37.5 degrees.  The chord length is x.  Then from the law of sines,

 

x / sin(105) = 1 / sin(37.5)

 

x = sin(105) / sin(37.5) ≅ 1.59 units

Answer 3

P=(cos45°, sin45°) = (√2/2, √2/2)

 

Q =(cos150°,sin150°) = (-√3/2, 1/2)

 

Length of chord = √([-√3/2 - √2/2]² + [1/2 - √2/2]²)

 

                       = √(0.75 + √6/2 + 0.5 + 0.25 - √2/2 + 0.5)

 

                       = √(√6/2 - √2/2 + 2)