Question

in the diagram, m is the midpoint of the segment. find the indicated length. 5. find ln. 6. find am. 7. find mr. diagrams: segment l-m-n with lm = x + 9, mn = 4x; segment a-m-c with am = x + 20, mc = 5x - 4; segment p-m-r with pm = 4x - 12, mr = -2x + 21

Explanation:

Problem 5: Find \( LN \)

Step 1: Set \( LM = MN \) (Midpoint Definition)

Since \( M \) is the midpoint, \( LM = MN \). So, \( x + 9 = 4x \).

Step 2: Solve for \( x \)

Subtract \( x \) from both sides: \( 9 = 3x \). Then divide by 3: \( x = 3 \).

Step 3: Find \( LN = LM + MN \)

\( LM = x + 9 = 3 + 9 = 12 \), \( MN = 4x = 4\times3 = 12 \). So \( LN = 12 + 12 = 24 \).

Step 1: Set \( AM = MC \) (Midpoint Definition)

\( x + 20 = 5x - 4 \).

Step 2: Solve for \( x \)

Subtract \( x \) and add 4: \( 24 = 4x \). Divide by 4: \( x = 6 \).

Step 3: Find \( AM \)

\( AM = x + 20 = 6 + 20 = 26 \).

Step 1: Set \( PM = MR \)

\( 4x - 12 = -2x + 21 \).

Step 2: Solve for \( x \)

Add \( 2x \) and 12: \( 6x = 33 \). Divide by 6: \( x = \frac{33}{6} = \frac{11}{2} = 5.5 \).

Step 3: Find \( MR \)

\( MR = -2x + 21 = -2\times5.5 + 21 = -11 + 21 = 10 \). (Or \( PM = 4x - 12 = 4\times5.5 - 12 = 22 - 12 = 10 \), same as \( MR \))

Answer:

\( 24 \)

Problem 6: Find \( AM \)