Problem Solving

[Math Revolution GMAT math practice question]

Points A(1,4), B(2,2) and C(p,1) lie on the x-y coordinate plane. If lines AB and BC are perpendicular to each other, p=?

A. -2
B. -1
C. 0
D. 1
E. 2

Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

Points A(1,4), B(2,2) and C(p,1) lie on the x-y coordinate plane. If lines AB and BC are perpendicular to each other, p=?

A. -2
B. -1
C. 0
D. 1
E. 2

$$?\,\, = \,p$$
Line AB is oblique (non-horizontal and non-vertical) therefore line BC is also oblique, and the product of their slopes must be -1.
$${\text{slope}}{\,_{\overleftrightarrow {{\text{AB}}}}}\,\, = \,\,\,\frac{{4 - 2}}{{1 - 2}}\,\, = \, - 2\,$$
$$\frac{1}{2}\,\,{\text{ = }}\,\,{\text{slope}}{\,_{\overleftrightarrow {{\text{BC}}}}}\,\,\,\,{\text{ = }}\,\,\frac{{2 - 1}}{{2 - p}}\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,p = 0$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

=>

The slope of the line segment joining the two points (x1,y1) and (x2,y2) is (y1-y2)/(x1-x2). The product of the slopes of perpendicular lines is -1.

The slope of the line AB is (4-2)/(1-2) = -2.
Since the two lines AB and BC are perpendicular each other, the slope of BC, which is (2-1)/(2-p), must be 1/2.
So,
(2-1)/(2-p)=1/2
1/(2-p)=1/2.
Thus, 2 - p = 2, and p = 0.

Therefore, the answer is C.
Answer: C