. If  LK MK, LK = 7x – 10, KN = x + 3, MN = 9x – 11, and KJ = 28, find LJ.

Let's break down the information given in the problem:

- \( LK = MK \)
- \( LK = 7x - 10 \)
- \( KN = x + 3 \)
- \( MN = 9x - 11 \)
- \( KJ = 28 \)
  
The goal is to find the length of \( LJ \).

### Step 1: Use the fact that \( LK = MK \)
Since \( LK = MK \), we know that:

\[ LK + KN = MN \]

Substitute the expressions for \( LK \), \( KN \), and \( MN \):

\[ (7x - 10) + (x + 3) = 9x - 11 \]

### Step 2: Simplify the equation
Combine like terms on the left-hand side:

\[
8x - 7 = 9x - 11
\]

Now, solve for \( x \):

### Step 3: Solve for \( x \)
Subtract \( 8x \) from both sides:

\[
-7 = x - 11
\]

Add 11 to both sides:

\[
4 = x
\]

### Step 4: Substitute \( x = 4 \) back into the expressions
Now, substitute \( x = 4 \) into the equations for \( LK \), \( KN \), and \( MN \):

- \( LK = 7(4) - 10 = 28 - 10 = 18 \)
- \( KN = 4 + 3 = 7 \)
- \( MN = 9(4) - 11 = 36 - 11 = 25 \)

### Step 5: Find the total length of \( LJ \)
To find \( LJ \), we need to sum the lengths of \( LK \), \( KJ \), and \( KN \):

\[
LJ = LK + KJ + KN = 18 + 28 + 7 = 53
\]

Thus, the length of \( LJ \) is:

\[
\boxed{53}
\]

Would you like more details on any part of this solution or have any questions?

Here are some related questions to expand on the topic:
1. What if \( KJ \) was not given? Could we find it using other information?
2. How would the problem change if \( LK \neq MK \)?
3. Can we find \( MJ \) given that \( MN = 25 \)?
4. What is the importance of knowing that \( LK = MK \)?
5. What types of geometric figures might this represent?

**Tip:** Always remember to check for consistency in given conditions when solving equations involving segments.

Learn how to solve a geometry problem involving segment lengths LK, KN, MN, and KJ to find the length of LJ. Detailed step-by-step solution provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation