Question 695595: How do you find the measures of the angles of a triangle when the angles are in the ratio of 6:8:10?
Found 2 solutions by RedemptiveMath, MathTherapy:
Answer by RedemptiveMath(80)   (Show Source):
You can put this solution on YOUR website! First we need to identify what the sum of the measures of the interior angles of a triangle are. We know that number to be 180°. We know that the three interior angles must add up to 180°. However, they must also be in the ratio 6:8:10. We could simply use x as the common factor of each angle. That is, we have 6x as the first angle, 8x as the second and 10x as the third. Now we can solve for x and then the angles:
6x + 8x + 10x = 180°
24x = 180°
x = 7.5.
Therefore, angle 1 = 6 * 7.5 or 45°, angle 2 = 8 * 7.5 or 60°, and angle 3 = 10 * 7.5 or 75°. We can check by seeing if they match the ratio 6:8:10. We must understand that 6:8:10 is a compound ratio. That is, we are dealing with a ratio 6:8, a ratio 8:10 and a ratio 6:10. The respective numbers must match these ratios:
45/60 = 9/12 = 6/8 (45 must correspond with 6 because 45 was 6x; 60 with 8x).
60/75 = 12/15 = 8/10 (75 must correspond with 10 because 75 was 10x).
45/75 = 9/15 = 6/10.
45/60 = 6/8 or 6:8. 60/75 = 8/10 or 8:10. 45/75 = 6/10 or 6:10. If we put these into a compound ratio, we'd have 6:8:10. We have proven that the angle measures 45°, 60° and 75° work for this problem. The reason we can simply use x with the ratio numbers and add them up to 180° can be explained below.
Looking at our problem, we can notice that there are common pieces between the numbers. These pieces are namely the ratio or fractions from the ratio we used in the problem. The reason we could simply use x is because that since this compound ratio must be true no matter what the values of the angles are, we'd get this ratio by using a common factor x. The common factor between 45, 60 and 75 is 7.5, which this was our x. Therefore, we can use the variable x to describe the ratio 6x:8x:10x.
But you may ask, "How come I can reduce the fractions most of the fractions mentioned in this answer?" The answer is simple. You can. In fact, it might save you some time if you do. 6:8:10 can be reduced to the ratio 3:4:5. The fractions 6/8, 8/10 and 6/10 can be reduced to 3/4, 4/5, and 3/5. I wouldn't reduce ratios if we were dealing with scale factors, measurements, magnification, and/or reduction, but this case is safe. If we had used 3x:4x:5x, we'd get x = 15 as our common factor. This would not matter because we'd still get 45°, 60° and 75° as our three angles. X can be any number as long as a ratio equivalent to 6:8:10 is held when x is solved for.
Answer by MathTherapy(10555)  (Show Source):
|
|
|
