In ΔABC , m ∠A = 4x, m ∠B = 3x+1, m ∠C = 7x+11. List the sides of ΔABC in order from shortest to longest. (1 point) Responses ;; Image with alt text: line AB ; Image with alt text: line BC ; Image with alt text: line AC ;; Image with alt text: line AB ; Image with alt text: line AC ; Image with alt text: line BC ;; Image with alt text: line BC ; Image with alt text: line AC ; ; ;
To find the order of the sides from shortest to longest, we need to compare the angles and their measurements.
Given that m ∠A = 4x, m ∠B = 3x+1, and m ∠C = 7x+11, we can compare these angles as follows:
4x < 3x+1 < 7x+11
To determine the order of the sides, we need to first solve for x.
Let's solve the above inequality:
4x < 3x+1 < 7x+11
Subtract 4x from all terms:
0 < -x+1 < 3x+11
Now, subtract 1 from all terms:
-1 < -x < 3x+10
Multiply all terms by -1 (Remember when we multiply an inequality by a negative number, we flip the signs):
1 > x > -3x-10
Divide all terms by -3 (Remember when we divide an inequality by a negative number, we flip the signs):
1/-3 < x/-3 > (-3x-10)/-3
-1/3 > x > -3x/3 - 10/3
Simplifying:
-1/3 > x > -x - 10/3
Now, let's examine the intervals:
-1/3 > x: Angle A is negative, which is not possible in a triangle, so this interval is not valid.
x > -x - 10/3
Simplifying:
2x > -10/3
Divide by 2:
x > -5/3
So, the valid interval for x is: x > -5/3
Now, let's substitute the values of x into the angle measurements to determine their sizes:
m ∠A = 4x = 4(-5/3) = -20/3 (not valid since angle measures cannot be negative)
m ∠B = 3x+1 = 3(-5/3)+1 = -5+1 = -4 (not valid since angle measures cannot be negative)
Therefore, the given measurements of the angles are not valid since they are negative.
Hence, it is not possible to determine the order of the sides from shortest to longest based on the provided angle measurements.