two angles ina triangle measure 24 degrees and 38 degrees. the longest side is 24cm long than the shortest side. how long is the shortest side?
To find the length of the shortest side of the triangle, we can use the fact that the sum of the angles in a triangle is always 180 degrees.
Let x be the measure of the third angle. We know that the sum of the three angles is 180 degrees:
24 + 38 + x = 180
Now, we can solve for x:
x = 180 - (24 + 38)
x = 118
So, the third angle measures 118 degrees.
Now, we can use the Law of Sines to find the lengths of the sides of the triangle. The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of the opposite angle is constant.
Let's denote the length of the shortest side as a, the length of the longest side as b, and the measure of the angle opposite the shortest side as A. We can set up the following equation:
a / sin(A) = b / sin(B)
In this case, we know that angle A measures 24 degrees and angle B measures 38 degrees. We can substitute these values into the equation:
a / sin(24) = (a + 24) / sin(38)
Now, we can solve for a:
a = (sin(24) * (a + 24)) / sin(38)
To simplify the equation, we can rearrange it:
a * sin(38) = sin(24) * (a + 24)
Using a scientific calculator or a sine table, we can find the values of sin(38) and sin(24). Then, we can solve the equation for a:
a * 0.6157 = 0.4077 * (a + 24)
0.6157a = 0.4077a + 9.8256
0.2079a = 9.8256
a = 47.24
Therefore, the shortest side of the triangle is approximately 47.24 cm long.
The angles are A= 24°, B=38°, C=118°, and the corresponding sides (of increasing length) are a, b and c.
Use sine rule to get:
(c+24)/sinA = c/sinC
Solve for c.