Question

if lm = 7 and ln = 12, which of the following statements must be true?
a. if points l, m, and n are collinear, then mn = 5.

b. if mn = 19, then points l, m, and n cannot be collinear.

c. if points l, m, and n are collinear, then mn must be either 5 or 19

d. it is not possible for points l, m, and n to be collinear.

Explanation:

Step1: Recall Collinear Points Property

When three points \( L \), \( M \), \( N \) are collinear, the length of the segment between the two outer points is the sum or difference of the lengths of the two adjacent segments, depending on the order of the points. So, we have two cases:

• Case 1: \( M \) lies between \( L \) and \( N \). Then \( LN = LM + MN \), so \( MN = LN - LM \).
• Case 2: \( L \) lies between \( M \) and \( N \). Then \( MN = LM + LN \).

Step2: Analyze Each Option

• Option A: If \( L \), \( M \), \( N \) are collinear, \( MN \) could be \( 5 \) (if \( M \) is between \( L \) and \( N \)) or \( 19 \) (if \( L \) is between \( M \) and \( N \)). So it's not necessarily \( 5 \). Eliminate A.
• Option B: If \( MN = 19 \), then using the collinear property, \( MN = LM + LN = 7 + 12 = 19 \), which means \( L \) is between \( M \) and \( N \), so they can be collinear. Eliminate B.
• Option C: From the two cases above, when \( L \), \( M \), \( N \) are collinear, either \( MN = LN - LM = 12 - 7 = 5 \) (if \( M \) is between \( L \) and \( N \)) or \( MN = LM + LN = 7 + 12 = 19 \) (if \( L \) is between \( M \) and \( N \)). So \( MN \) must be either \( 5 \) or \( 19 \). This is correct.
• Option D: As shown in the cases, the points can be collinear. Eliminate D.

Answer:

C. If points \( L \), \( M \), and \( N \) are collinear, then \( MN \) must be either \( 5 \) or \( 19 \)