Question

In a square ABCD, A(1,3) and C(4,2). AC is a diagonal. Express the coordinates of a point on the diagonal BD using a real parameter. Hence find the coordinates of the other two vertices.

Answer 1

Answer on Question #83981 – Math – Analytic Geometry

Question

In a square $ABCD$, $A(1,3)$ and $C(4,2)$ are two vertices. AC is a diagonal. Express the coordinates of a point on the diagonal BD using a real parameter. Hence find the coordinates of the other two vertices.

Solution

The point $M\left(\frac{A_x + C_x}{2}, \frac{A_y + C_y}{2}\right) = M(2.5, 2.5)$ is the center of the square.

The parametric equations of $AC$ are $x = M_x + (C_x - A_x)t$, $y = M_y + (C_y - A_y)t$.

The parametric equations of $BD$ are $x = M_x - (C_y - A_y)s$, $y = M_y + (C_x - A_x)s$.

There are relationships between vertexes:

$$\begin{array}{l} \overrightarrow{r_{BD}} = \overrightarrow{r_{AC}} \\ \overrightarrow{r_{OA}} = \overrightarrow{r_{OM}} + t_A \overrightarrow{r_{AC}} \\ \overrightarrow{r_{OC}} = \overrightarrow{r_{OM}} + t_C \overrightarrow{r_{AC}} = \overrightarrow{r_{OM}} - t_A \overrightarrow{r_{AC}}, \\ \overrightarrow{r_{OB}} = \overrightarrow{r_{OM}} + s_B \overrightarrow{r_{BD}} = \overrightarrow{r_{OM}} + t_A \overrightarrow{r_{BD}}, \\ \overrightarrow{r_{OD}} = \overrightarrow{r_{OM}} + s_D \overrightarrow{r_{BD}} = \overrightarrow{r_{OM}} - s_B \overrightarrow{r_{BD}} \\ \end{array}$$

The coordinates of the vertex B are

$$x = M_x - (C_y - A_y) s_B = M_x - (C_y - A_y) t_A = M_x - \frac{(C_y - A_y)(A_x - M_x)}{(C_x - A_x)} = 2.5 - \frac{-1*(-1.5)}{3} = 2,$$ $$y = M_y + (C_x - A_x) s_B = M_x + (C_x - A_x) t_A = M_y + \frac{(C_x - A_x)(A_x - M_x)}{(C_x - A_x)} = 2.5 + \frac{3*(-1.5)}{3} = 1$$

The coordinates of the vertex D are

$$x = M_x - (C_y - A_y) s_D = M_x + (C_y - A_y) s_B = 2.5 + \frac{-1*(-1.5)}{3} = 3,$$ $$y = M_y + (C_x - A_x) s_D = M_y - (C_x - A_x) s_B = 2.5 - \frac{3*(-1.5)}{3} = 4$$

Answer:

The other two vertices are $B(2,1)$ and $D(3,4)$.

Answer provided by https://www.AssignmentExpert.com