How you know when there is only one solution triangle possible from a side-side-angle?
1 answer:
Answer:
There will be only on solution if either of these conditions is met:
- the given angle is opposite the longest side
- the triangle is a right triangle (the ratio of the side opposite the angle to the other given side is equal to the sine of the angle)
Step-by-step explanation:
Consider the SSA geometry with the angle placed in standard position at the origin and its adjacent given side (the second side of SSA) extending along the +x axis. Draw a circle having a radius equal to the length of the first side of SSA, centered on the vertex between the two segments on the +x axis.
There are three possibilities for the way this circle intersects the other ray of the angle (the ray that is not the x-axis):
- the first side is as long or longer than the second side, so there is one point of intersection (one solution to the triangle, purple in the attachment)
- the first side is just long enough to be tangent to the other ray, so there is one point of intersection (one solution to the triangle, green in the attachment)
- the first side is between these lengths, so intersects the other ray in two places, giving two solutions to the triangle, red in the attachment.
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