Question

1. if b is the mid - point of ac, ac = cd, ab = 3x + 4, ac = 11x - 17, and ce = 49, find

Explanation:

Step1: Use mid - point property

Since $B$ is the mid - point of $AC$, then $AB=\frac{1}{2}AC$. So $2AB = AC$. Given $AB = 3x + 4$ and $AC=11x - 17$, we have $2(3x + 4)=11x - 17$.
Expand the left - hand side: $6x+8 = 11x - 17$.

Step2: Solve for $x$

Subtract $6x$ from both sides: $8=11x - 6x-17$.
Simplify to get $8 = 5x-17$.
Add 17 to both sides: $8 + 17=5x$, so $25 = 5x$.
Divide both sides by 5: $x = 5$.

Step3: Find the length of $AC$

Substitute $x = 5$ into the formula for $AC$: $AC=11x - 17$.
$AC=11\times5-17=55 - 17=38$.
Since $AC = CD$, then $CD = 38$.

Step4: Find the length of $DE$

We know that $CE = 49$ and $CD = 38$.
Since $CE=CD + DE$, then $DE=CE - CD$.
$DE=49 - 38 = 11$.

Answer:

$DE = 11$