Question

A rectangle has two opposite vertices at the points $(1,2)$ and $(5,5)$. If the other vertices lie on the line $x=3$, then the coordinates of the other vertices are

• A. $(3,-1),(3,-6)$
• B. $(3,1),(3,5)$
• C. $(3,2),(3,6)$
• D. $(3,1),(3,6)$

Answer 1

Answer: (D)

$(3,1),(3,6)$

Let $A\equiv (1,2)$ and $C\equiv (5,5)$.

Since the vertices $B$ and $D$ lie on the line $x=3$, therefore, let $B\equiv (3,y_1)$ and $D\equiv (3,y_2)$.

Since $AC$ and $BD$ bisect each other, so they have same middle point

$\Rightarrow \frac{y_1+y_2}{2}=\frac{2+5}{2}\Rightarrow y_1+y_2=7$ ...(1)

Distance Formula $AC=\sqrt{{(x_1-x_2)}^2+{(y_1-y_2)}^2}$

Also ${BD}^2={AC}^2\Rightarrow {(y_1-y_2)}^2={(1-5)}^2+{(2-5)}^2=25\Rightarrow y_1-y_2=5$ ...(2)

Subtracting (1) from (2) we get

$y_2=6$

$\therefore y_1=1$

Thus, the other vertices of the rectangle are $(3,1)$ and $(3,6)$.