One end point of the line segment $P_1P_2$ has coordinates $P_1(-1, 2, 5)$

Question

Question:

The midpoint $P$ is known to lie in the $xz-$plane while the other end point is known to lie on the intersection of the planes $x=5$ and $z=8$.

1. Find $P$ and $P_2$

2. Sketch all the points and the planes

My Try:

Suppose $P(x_p, 0, y_p)$ be a midpoint on the $xz-$plane and $P_2(5, y_{p_2}, 8)$ be an point lies on the intersection of the planes $x=5$ and $z=8$. Then we know

$|PP_1|=|PP_2|$

Is this correct approach to find $P$ and $P_2$?

Pls suggest me a way or provide me the solution for this problem. Thanks.

Answer 1

We can write the coordinate of three points as follows:

$P_1(-1, 2, 5)$, $P(x_P, 0, z_P)$ and $P_2(5, y_{P_2}, 8)$

We must have:

$$x_P=\frac{-1+5}2=2$$

$$y_P=\frac{2+y_{P_2}}2=1+\frac {y_{P_2}}2=0\Rightarrow y_{P_2}=-2$$

$$z_P=\frac{5+8}2=\frac {13}2$$

In this way : $P(2, 0, \frac{13}2)$ and $P_2(5, -2, 8)$.