In (triangle)EFG, if m<E = 6x - 8, m<F = 7x + 3, and m<G = 3x - 7. State the sides correctly ordered from longest to shortest.
To determine the correct order of the sides from longest to shortest in triangle EFG, we need to compare the lengths expressed by the given equations.
Given:
m<E = 6x - 8
m<F = 7x + 3
m<G = 3x - 7
To compare the lengths, we can compare the expressions.
The side lengths in this case are represented by 6x - 8, 7x + 3, and 3x - 7.
To order the side lengths from longest to shortest, we will compare the coefficients (numbers in front of x) and constants (numbers without x) of each expression.
Comparing the coefficients:
6 (from 6x) < 7 (from 7x) < 3 (from 3x)
Comparing the constants:
-8 (from -8) < 3 (from +3) < -7 (from -7)
From the above comparisons, we can conclude that the correct order of the sides from longest to shortest is:
Side F > Side G > Side E
To determine the correct order of the sides in triangle EFG from longest to shortest, we need to compare the angles formed by the sides. The lengths of the sides are not given directly in the question, but we can use the angles to make a determination.
In a triangle, the largest side is opposite the largest angle, the second-largest side is opposite the second-largest angle, and the smallest side is opposite the smallest angle.
Given that m<E = 6x - 8, m<F = 7x + 3, and m<G = 3x - 7, we can equate these angles to determine their relationships:
m<E = 6x - 8
m<F = 7x + 3
m<G = 3x - 7
To determine the longest side, we need to find the largest angle.
Comparing the given angles, we have:
6x - 8 < 7x + 3
-8 < x + 3
-11 < x
This means that x is greater than -11.
Next, we compare the remaining angles:
7x + 3 < 3x - 7
7x - 3x < -7 - 3
4x < -10
x < -10/4
x < -5/2
This means that x is less than -5/2.
So, by deducing the range of x, we can determine the order of the angles.
-11 < x < -5/2
Since the angles are sorted in the form EFG, the correct order of the sides from longest to shortest is: FG, EF, GE.
6x-8 + 7x+3 + 3x-7 = 180
x = 12
So, evaluate for each angle, then sort them.