Question

If $P$ is any point within a $∆ABC,$ then $\mathrm{PA}+\mathrm{CP}$ is equal to?

• A. $AC+CB$
• B. $BC+BA$
• C. $CB+AB$
• D. $CB+BA$

Answer 1

The correct option is D

$CB+BA$

Explanation for the correct options:

Step 1: Draw a required triangle

$∆ABC$ shown in the figure

Here, $P$ is the mid point or center of the triangle and $AB,BA,\mathrm{and}CA$ are the sides of the triangle.

The triangle property states that the third side of a triangle is always equal to the sum of the lengths of any two of its sides.

Therefore,$\overrightarrow{CP}+\overrightarrow{PA}=\overrightarrow{CA}....\left(1\right)$

$\overrightarrow{CA}=\overrightarrow{CB}+\overrightarrow{BA}$ [By Triangle Laws]

Step 2: Substituting the value of $\overrightarrow{CA}$ in equation (1) we get,

$$\begin{gathered} \Rightarrow \overrightarrow{\mathrm{CP}}+\overrightarrow{\mathrm{PA}}=\overrightarrow{\mathrm{CB}}+\overrightarrow{\mathrm{BA}} \\ \Rightarrow \overrightarrow{\mathrm{PA}}+\overrightarrow{\mathrm{CP}}=\overrightarrow{\mathrm{CB}}+\overrightarrow{\mathrm{BA}} \end{gathered}$$

Hence, the correct answer is an option (D)