CBSE BoardEnglish MediumSTD 10MathsCoordinate Geometry1 Mark
MCQ
If points $A(5, p), B(1, 5), C(2, 1)$ and $D(6, 2)$ form a square $\text{ABCD},$ then $p =$
A
$7$
B
$3$
✓
$6$
D
$8$
✓
Answer
Correct option: C.
$6$
The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula,
$\text{d}=\sqrt{(\text{x}_1-\text{x}_2)^2+(\text{y}_1-\text{y}_2)^2}$
In a square all the sides are equal to each other.
Here the four points are $A(5, p), B(1, 5), C(2, 1)$ and $D(6, 2).$
The vertex $'A\ '$ should be equidistant from $'B\ '$ as well as $'D\ '.$
Let us now find out the distance $'AB\ '$ and $'AD\ '.$
$\text{AB}=\sqrt{(5-1)^2+(\text{p}-5)^2}$
$\text{AB}=\sqrt{(4)^2+(\text{p}-5)^2}$
$\text{AD}=\sqrt{(5-6)^2+(\text{p}-2)^2}$
$\text{AD}=\sqrt{(-1)^2+(\text{p}-2)^2}$
These two need to be equal.
Equating the above two equations we have,
$AB = AD$
$\sqrt{(4)^2+(\text{p}-5)^2}$
$=\sqrt{(-1)^2+(\text{p}-2)^2}$
Squaring on both sides we have,
$(4)^2+(p-5)^2=(-1)^2+(p-2)^2 $
$ 16+p^2+25-10 p$
$=1+p^2+4-4 p $
$6p = 36$
$p = 6$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.