Answer:
- y = (2+√3)x -(1+2√3)
- y = (2 -√3)x -(1-2√3)
Step-by-step explanation:
You want the equations of the other two sides of an equilateral triangle with vertex (2, 3) and opposite side x +y = 2.
Slope
The slope of the given line is -1, so it makes an angle of 135° with the +x axis. The other sides of the triangle will make angles that differ from this by multiples of 60°. This means the slopes of the other two sides will be ...
Point-slope equation
The point-slope equations of the other two lines will be of the form ...
y -k = m(x -h) . . . . . . . . line with slope m through point (h, k)
Each of the missing sides will include the vertex (2, 3), so the equations can be written ...
y -3 = tan(75°)(x -2)
y -3 = tan(15°)(x -2)
Using the exact values of these tangents, we find the slope-intercept equations to be ...
- y = (2+√3)x -(1+2√3)
- y = (2 -√3)x -(1-2√3)
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Additional comment
The exact values of these tangents can be found using the sum and difference formulas for the tangent, along with tan(45°) = 1 and tan(30°) = √3/3.
tan(a+b) = (tan(a) +tan(b))/(1 -tan(a)tan(b)) ⇒ tan(75°) = 2+√3
tan(a-b) = (tan(a) -tan(b))/(1 +tan(a)tan(b)) ⇒ tan(15°) = 2-√3
