In triangle ABC m<A=4x, m<B=3x+1, m<C=7x+11. List the sides of triangle ABC in order from shortest to longest
To determine the side lengths of the triangle, we need to know the relationship between the angles and the sides. If we assume that these are the degrees of the angles (as opposed to radians), then we can use the triangle inequality theorem to determine the side order.
The triangle inequality theorem states that in a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Let's analyze the given angles:
m<A = 4x
m<B = 3x + 1
m<C = 7x + 11
Now, let's assume that the smallest angle is angle A. In this case, its side length should be the smallest. Let's analyze the possible cases:
Case 1: A < B < C
4x < 3x + 1 < 7x + 11
Simplifying these inequalities:
4x < 3x + 1,
3x < 7x + 11
Subtracting 3x from both sides of the first inequality will give us:
x < 1,
0 < 4x + 11
From the second inequality, we can conclude that x > -11/4. Combining these results, we know that x must be between -11/4 and 1. In this case, A is the smallest angle, and we can order the sides in increasing order.
Case 2: B < A < C
3x + 1 < 4x < 7x + 11
Similarly simplifying:
3x + 1 < 4x,
4x < 7x + 11
From the first inequality, subtracting 3x gives us 1 < x. From the second inequality, subtracting 4x gives us 0 < 3x + 11.
Combining these inequalities, we see that x must be greater than 1. In this case, B is the smallest angle, and we can order the sides accordingly.
Case 3: B < C < A
3x + 1 < 7x + 11 < 4x
Analogously simplifying:
3x + 1 < 7x + 11,
7x + 11 < 4x
From the first inequality, subtracting 3x and 11 on both sides gives us -10 < 4x. From the second inequality, subtracting 7x gives us 11 < -3x.
However, this case leads to a contradiction because for the inequality 11 < -3x to hold true, x would have to be negative. However, in this context, x represents an angle, and degrees cannot be negative. Therefore, this case is not valid.
Therefore, we can conclude that the correct order of the sides from shortest to longest will depend on the values of x. If x is between -11/4 and 1, the shortest side will correspond to angle A; if x is greater than 1, the shortest side will correspond to angle B.