# Finding the location of points of a triangle given the angle and length ratio.

Given that a point P is located at (-2.5,4.33) I need to locate the points A and B such that $$\frac{PA}{PB} = \frac{4.77}{8}$$ and $$\angle APB = 55^o$$.

A and B must be on the -ve part of the x axis.

I need to get A and B located in order to solve a problem in Control Systems Lag-Lead Compensator design. The problem is found in Modern Control Engineering by Ogata.

• Isn't there any other condition or information? The problem as written has infinity solutions (A,B) – user376343 Dec 2 '18 at 9:45
• That information alone is not enough. You can rotate the triangle around point $P$ however you like and get infinitely many different values for $A$ and $B$. Or are you looking for a general expression? – platty Dec 2 '18 at 9:46
• Sorry my bad you both are right I forgot to mention that A and B must be on the x-axis (-ve part of x-axis to be specific) – Hasan Hammoud Dec 2 '18 at 9:48

HINT: Let $$A= (x_a , 0)$$ and $$B=(x_b, 0)$$, with $$x_a < x_b < 0$$. You need to find $$x_a$$ and $$x_b$$. Consider a system of two equations:
The second equation can be written using the Law of cosine, (see https://en.wikipedia.org/wiki/Law_of_cosines), since you know the cosine of the angle $$\angle APB$$.