Let's solve the problems from the image step by step.

Problem 7:

We are given three equations:

• $BD = 7x - 10$
• $BC = 4x - 29$
• $CD = 5x - 9$

Since $BD = BC + CD$ (because $B$, $C$, and $D$ are collinear points), we can set up the following equation:

$BD = BC + CD$

Substituting the given values:

$7x - 10 = (4x - 29) + (5x - 9)$

Now, let's solve for $x$:

$7x - 10 = 4x - 29 + 5x - 9$

Combine like terms:

$7x - 10 = 9x - 38$

Subtract $9x$ from both sides:

$7x - 9x - 10 = -38$

$-2x - 10 = -38$

Add 10 to both sides:

$-2x = -28$

Divide by -2:

$x = 14$

Now that we have $x = 14$, let's find the values for $BC$, $CD$, and $BD$:

1. $BC = 4x - 29 = 4(14) - 29 = 56 - 29 = 27$
2. $CD = 5x - 9 = 5(14) - 9 = 70 - 9 = 61$
3. $BD = 7x - 10 = 7(14) - 10 = 98 - 10 = 88$

Final answers for Problem 7:

• $x = 14$
• $BC = 27$
• $CD = 61$
• $BD = 88$

Problem 8:

We are given the following conditions:

• $BD \equiv BC$ (i.e., $BD = BC$)
• $BD = 5x - 26$
• $BC = 2x + 1$
• $AC = 43$

Since $BD = BC$, we can set the two expressions equal:

$5x - 26 = 2x + 1$

Now, solve for $x$:

$5x - 2x = 1 + 26$

$3x = 27$

$x = 9$

Now that we have $x = 9$, substitute this into either $BD$ or $BC$ (since they are equal):

$BD = 5(9) - 26 = 45 - 26 = 19$

Next, we know that $AB + BC = AC$, so:

$AB + 19 = 43$

Solve for $AB$:

$AB = 43 - 19 = 24$

Final answer for Problem 8:

• $AB = 24$

Would you like further clarification or more details on any of these solutions?

Here are five related questions to practice with:

1. If $BD = 6x - 12$ and $BC = 3x - 8$, and $CD = 4x - 5$, how would you solve for $x$?
2. What if $BD \neq BC$? How would that change the solution in problem 8?
3. How can you verify that a line segment is congruent to another?
4. Can you apply these steps to solve problems involving triangles instead of line segments?
5. What are some real-life applications of these geometric principles?

Tip: Always set up relationships between segments or angles before solving equations to ensure accuracy.