Question

If the point P lies in between M \& N and C is the mid point of MP then which of the following option is correct :

• A. MC + PN = MN
• B. MP + CP = MN
• C. MC + CP = MN
• D. CP + CN = MN

Hint:

1. Draw a line segment and place the points M, C, P, N in order based on the problem statement: M --- C --- P --- N 2. From "P lies between M and N": \(MP + PN = MN\). 3. From "C is the midpoint of MP": \(MC = CP\) and \(MP = 2CP = 2MC\). 4. Test Option (4): \(CP + CN = MN\). The segment CN is composed of CP and PN. So, \(CN = CP + PN\). Substitute this into the option: \(CP + (CP + PN) = MN\). This simplifies to \(2CP + PN = MN\). Since \(MP = 2CP\), this becomes \(MP + PN = MN\), which is true from point 2. Thus, option (4) is correct.

Answer 1

The Correct Option is D

Concept: This problem involves understanding segment addition and midpoint properties on a line segment. Step 1: Understand the given information about the points (A) Point P lies in between M and N. This means M, P, and N are collinear, and P is somewhere on the segment MN. This implies the segment addition postulate: \[ MP + PN = MN \quad \quad (*)\ \] (B) C is the midpoint of the segment MP. This means C lies on MP and divides MP into two equal parts: \[ MC = CP \] Also, \(MP = MC + CP\). Since \(MC = CP\), we have \(MP = CP + CP = 2CP\). And \(MP = MC + MC = 2MC\). From these, the order of points on the line segment is M-C-P-N. Step 2: Analyze the target equation (Option 4) We want to check if \(CP + CN = MN\) is correct. From the order of points M-C-P-N, we can see that point P lies between C and N. Therefore, by the segment addition postulate applied to segment CN with point P in between: \[ CP + PN = CN \quad \quad (**)\ \] This relationship is true based on the order M-C-P-N. Now let's try to relate this to MN. We know from (*) that \(MN = MP + PN\). We also know from the midpoint property that \(MP = 2CP\). Substitute \(MP = 2CP\) into (*): \[ MN = 2CP + PN \] The target option is \(CP + CN = MN\). Let's substitute \(CN = CP + PN\) (from **) into the target option: LHS of Option (4) = \(CP + CN = CP + (CP + PN) = 2CP + PN\). RHS of Option (4) = \(MN\). Since we derived \(MN = 2CP + PN\), then LHS = RHS. So, \(CP + CN = MN\) is correct. Alternative approach for Step 2: Directly verify each option Option (1): MC + PN = MN We know \(MC = CP\) and \(MN = MP + PN = 2CP + PN\). So, \(MC + PN = CP + PN\). Is \(CP + PN = 2CP + PN\)? This would mean \(0 = CP\), which is not generally true. So, Option (1) is incorrect unless P and C coincide (MP=0). Option (2): MP + CP = MN We know \(MN = MP + PN\). So, is \(MP + CP = MP + PN\)? This implies \(CP = PN\). This is not generally true. So, Option (2) is incorrect. Option (3): MC + CP = MN We know \(MC + CP = MP\). So, is \(MP = MN\)? This would mean \(PN = 0\), i.e., P and N coincide. This is not generally true. So, Option (3) is incorrect. Option (4): CP + CN = MN As established, the order of points is M-C-P-N. Segment CN consists of two parts: CP and PN. So, \(CN = CP + PN\). The equation becomes \(CP + (CP + PN) = MN\). This simplifies to \(2CP + PN = MN\). We know \(MP = 2CP\). So, the equation is \(MP + PN = MN\). This is true by the segment addition postulate since P is between M and N. Therefore, Option (4) is correct.