Question

given: the length of bd̅ exceeds the length of ce̅ by 10. 6. what is the length of cd̅? 9. which is longer, ac̅ or ce̅? 10. what is the total length of ae̅?

Explanation:

Step1: Set up an equation from the given condition

Given that the length of $\overline{BD}$ exceeds the length of $\overline{CE}$ by 10. The length of $\overline{BD}=(4x - 3)+3x=7x - 3$, and the length of $\overline{CE}=3x+(2x + 3)=5x+3$. So, $(7x - 3)-(5x + 3)=10$.

Step2: Simplify the left - hand side of the equation

Expand the left - hand side: $7x-3 - 5x - 3=10$. Combine like terms: $(7x-5x)+(-3 - 3)=10$, which gives $2x-6 = 10$.

Step3: Solve for x

Add 6 to both sides of the equation: $2x-6 + 6=10 + 6$, so $2x=16$. Divide both sides by 2: $x = 8$.

Step4: Find the length of $\overline{CD}$

The length of $\overline{CD}=3x$. Substitute $x = 8$ into the expression for the length of $\overline{CD}$, we get $3\times8 = 24$.

Step5: Find the lengths of $\overline{AC}$ and $\overline{CE}$

The length of $\overline{AC}=2x+(4x - 3)=6x - 3$. Substitute $x = 8$, then $\overline{AC}=6\times8-3=48 - 3=45$. The length of $\overline{CE}=5x + 3$. Substitute $x = 8$, then $\overline{CE}=5\times8+3=40 + 3=43$. Since $45>43$, $\overline{AC}$ is longer.

Step6: Find the length of $\overline{AE}$

The length of $\overline{AE}=2x+(4x - 3)+3x+(2x + 3)$. Combine like terms: $(2x+4x+3x+2x)+(-3 + 3)=11x$. Substitute $x = 8$, we get $11\times8=88$.

Answer:

1. The length of $\overline{CD}$ is 24.
2. $\overline{AC}$ is longer.
3. The total length of $\overline{AE}$ is 88.