i have two given co-ords and one angle of a triangle; i wanted to find the 3rd coord? how do i find it?

Question

the coords are A(229,41), B(229,206) and C(x,y)
AB=BC=165
angle ABC=45 degrees

Answer 1

There was not originally enough information to find the third coordinate. You would have been given the length of one side and the measure of one angle. Even if the problem specified which angle this is (adjacent to the given side, or opposite from it), there are infinitely many possible locations for the third coordinate. You can never uniquely define a triangle with fewer than three data points, and you were given only two (the coordinates of two points are a single piece of data with respect to triangle geometry because it defines a single side). You can use side-side-side (SSS), side-angle-side (SAS), or angle-side-angle (ASA). If you are given side-side-angle (SSA), it may uniquely definite a triangle or it may describe two different triangles, if it defines a triangle at all.

With your additional information, it can now be solved. You were actually given the lengths of two of the sides, because you were told that BC has length equal to that of AB, making this a SAS problem. Note that this is an isoceles triangle where you have been given the apex angle. The base angles of an isoceles triangle are congruent to each other, and the sum of all three angles is 180, so each base angle has measure (180 - B)/2, where B is the given apex angle.

You can find the length of AC by using the law of cosines. If the lengths of the sides of a triangle are a, b, and c, and the angle opposite side b has measure B (and so forth), then we can write b^2 = a^2 + c^2 - 2ac*cos(B). In this case, you have a = c = 165 and B = 45, so you can calculate b, the length of side AC.

If you don't already have an illustration of this problem, sketch one. Now draw a horizontal line that extends from point C to line AB. Lable the intersection as point D. You have a right triangle, ADC, where you just calculated the length of the hypotenuse, AC. You also know one of the angles in the right triangle, because it's the base angle of the original triangle. Simple trigonometry will enable you to calculate the lengths of AD (adjacent to the known angle) and DC (opposite the known angle). Since you know the coordinates of A, you just need to add the length of DC to the x-coordinate and add the length of AD to the y-coordinate to get the coordinates of C.

It should be noted that I assumed C lies to the right of the vertical line AB. In actuality, there is no reason why it cannot lie to the left instead. To be safe, you should solve it both ways. However, you don't need to do all the work again. The alternate point C' has the same y-coordinate as C, but the x-coordinate is found by subtracting the length of DC from the s-coordinate of A instead of adding it.