What is the relationship between angles in a triangle and the resulting area of the triangle?
1. How does the size of an angle in a given triangle relate to the Area of the triangle?
2. How does the sin of an angle relate to the area?
3. When graphing the area of a triangle against Sin(theta), a straight line is produced. What does the slope and equation of this line mean?
Any help greatly appreciated.
Thanks, John.
1)area=1/2a*b*sin(C) where; a,b,C are lengths &angle of usual tri angle that is A propotional to c
2) area=1/2a*b*sin(C)
3) 1/2*a*b area of rectangula triangle
1)area=1/2a*b*sin(C) where; a,b,C are lengths &angle of usual tri angle that is A propotional to c
2) area=1/2a*b*sin(C)
3) 1/2*a*b area of rectangula triangle
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I assume this means that the lengths of two sides are held constant, and you want to know how the area of the triangle changes when the angle between these two sides is varied. Call the two side lengths "a" and "b", and the angle between them C.
1. Here’s an intuitive argument, which the answer to part 2 will make formal. If the angle is very small, then you will end up with a skinny, tall triangle, so the area will be small. Similarly, if the angle is very large (that is, close to 180 degrees), then you end up a short, wide triangle, so the area is small again. The area is largest when C=90 degrees.
2. The formula for the area of the triangle is Area = 0.5ab*sin C. (It’s kind of a pain to add a diagram here, so I’ll let you try to prove this by yourself. The proof is very short. If you have trouble, you can ask again.)
3. If you look at the formula above, you can see that the slope of this line will be 0.5ab. This observation means that the area is proportional to sin C.
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In any triangle, the largest interior angle is opposite the largest side; the smallest interior angle is opposite the smallest side and, not surprisingly, the middle-sized interior angle is opposite the middle-sized side
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