Which of the following can be the ratio of the lengths of sides of a triangle: 4: 7: 8 and 3: 8: 13?
Celia measured the angles and the lengths of the sides of two triangles. She wrote down the ratios for the angle measures and the side lengths. Two of the ratios were 4: 7: 8 and 3: 8: 13. When Celia got to school the next day, she couldn't remember which ratio was for angles and which was for sides.
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The ratios that Celia had with her were 4: 7: 8 and 3: 8: 13.
Now for a triangle the sum of the lengths any two sides should be greater than the length of the third side. If this is not the case, the two sides do not meet and the triangle is not a closed figure.
Now, to test if the ratio of the lengths of sides is 4: 7: 8, let the length of the smallest side be 4L, the other two sides are 7L and 8L
We see that 4L + 7L = 11L > 8L
7L + 8L = 15L > 4L
8L + 4L = 12L > 7L
But when we test the ratio 3: 8: 13, and let the length of the smallest side be 3L, the other sides are 8L and 13L
Here 8L + 13L = 21L > 3L
13L + 3L = 16L > 8L
but 8L + 3L = 11L < 13L
Therefore the ratio 3: 8: 13 cannot be the ratio of the of the lengths of the sides.
Only 4: 7: 8 can be the ratio of the lengths of the sides.
Find the measures of the angles of one of Celia's triangles.
The given ratios are as follows:
4: 7: 8 and 3: 8: 13
The second ratio 3:8:13 given is not of sides as 3+8 < 13.
The reason is as follows:
In a triangle, the sum of the lengths of any two sides is greater than the 3rd side.
So even ratio of sides of the triangle should make a similar triangle. Therefore, by above consideration, if 3, 8 and 13 were the sides of a triangle, they do not obey the rule that the sum of any two sides > greater than the 3rd side.
Therefore 3:8:13 is not the ratio of sides:
Therefore 4: 7: 8 may be the ratio of the sides:
But even this does not pass the test of sine rule as shown below:
So we do the sine rule test.
Put a = 4, b= 7, c = 8 .
A= 180*3/(3+8+13) = 22.5 deg
B =180*/(3+8+13) = 60 deg
C = 180*13/(3+8+13) = 97.5 deg
A/sinA = b/sinB = c/sinC.
4/sin22.5 = 10.4525.
7/sin60 = 8.08.
13/sin97.5 = 8.07.
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