To solve this problem, we need to follow a series of steps very carefully. Let's start by analyzing and solving the given equations step-by-step.
Step 1: Determine [tex]\(x\)[/tex]
We know the following:
- [tex]\(AB = 3x + 4\)[/tex]
- [tex]\(AC = 11x - 17\)[/tex]
- [tex]\(B\)[/tex] is the midpoint of [tex]\(AC\)[/tex], which implies [tex]\(AB = BC\)[/tex].
By the midpoint property, we have:
[tex]\[ AB + BC = AC \][/tex]
Since [tex]\(B\)[/tex] is the midpoint of [tex]\(AC\)[/tex], [tex]\(AB = BC\)[/tex]. Hence:
[tex]\[ AB + AB = AC \][/tex]
[tex]\[ 2AB = AC \][/tex]
Substitute the given expressions for [tex]\(AB\)[/tex] and [tex]\(AC\)[/tex]:
[tex]\[ 2(3x + 4) = 11x - 17 \][/tex]
Step 2: Solve for [tex]\(x\)[/tex]
Distribute the 2 on the left side:
[tex]\[ 6x + 8 = 11x - 17 \][/tex]
Move all terms involving [tex]\(x\)[/tex] to one side and the constants to the other side:
[tex]\[ 6x - 11x = -17 - 8 \][/tex]
[tex]\[ -5x = -25 \][/tex]
Divide by [tex]\(-5\)[/tex]:
[tex]\[ x = \frac{-25}{-5} \][/tex]
[tex]\[ x = 5 \][/tex]
Step 3: Calculate [tex]\(AB\)[/tex] and [tex]\(AC\)[/tex]
Substitute [tex]\(x = 5\)[/tex] into the expressions for [tex]\(AB\)[/tex] and [tex]\(AC\)[/tex]:
[tex]\[ AB = 3(5) + 4 = 15 + 4 = 19 \][/tex]
[tex]\[ AC = 11(5) - 17 = 55 - 17 = 38 \][/tex]
Step 4: Confirm [tex]\(AC = CD\)[/tex]
Given [tex]\(AC = CD\)[/tex], we know:
[tex]\[ CD = 38 \][/tex]
Step 5: Find [tex]\(DE\)[/tex]
We are also given that [tex]\(CE = 49\)[/tex]. To find [tex]\(DE\)[/tex], we use the fact that [tex]\(CE\)[/tex] is composed of segments [tex]\(CD\)[/tex] and [tex]\(DE\)[/tex]:
[tex]\[ CE = CD + DE \][/tex]
[tex]\[ 49 = 38 + DE \][/tex]
Solving for [tex]\(DE\)[/tex]:
[tex]\[ DE = 49 - 38 = 11 \][/tex]
Final answers:
[tex]\[ x = 5 \][/tex]
[tex]\[ DE = 11 \][/tex]
So, the values are:
[tex]\[ x = 5 \][/tex]
[tex]\[ DE = 11 \][/tex]
