Carcass if The slopes of any two pair of points among the three points A, B and C are same then how come the slopes of AB and BC are such that they are perpendicular to each other

tyrant100


Sorry sir. I edited the OA because it was completely off.

See now if it is clear

Let me know if you need further assistance

C is not the answer as A, B, C are collinear. There AB in any case can't be perpendicular to BC

Let's analyze the points $\(A(4,6), B(8,10)\)$, and $\(C(12,14)\)$ :
A. Do points $\(\mathrm{A}, \mathrm{B}\)$, and C lie on the same straight line?

To check if three points lie on the same straight line (are collinear), we can check if the slopes between pairs of points are the same.
B. Are the slopes of any two pairs among the points $\(\mathrm{A}, \mathrm{B}\)$, and C the same?

Calculate slopes:
- Slope of $A B$ :

$$
\(m_{A B}=\frac{y_2-y_1}{x_2-x_1}=\frac{10-6}{8-4}=\frac{4}{4}=1\)
$$

- Slope of $B C$ :

$$
\(m_{B C}=\frac{14-10}{12-8}=\frac{4}{4}=1\)
$$

- Slope of $A C$ :

$$
\(m_{A C}=\frac{14-6}{12-4}=\frac{8}{8}=1\)
$$


Since all slopes are equal to 1 , the points are collinear.
Thus, both statements $A$ and $B$ are true.
C. Is line $A B$ perpendicular to line $B C$ ?

C. Is line $A B$ perpendicular to line $B C$ ?

Lines are perpendicular if the product of their slopes is -1 .
Check:

$$
\(m_{A B} \times m_{B C}=1 \times 1=1 \neq-1\)
$$


So, line $A B$ is not perpendicular to line $B C$.

Final evaluation:
- A. True
- B. True
- C. False