Question

*. If (2,3) and (-6,5) are the end points of diagonal of a square. Find the equation of another diagonal.

Answer 1

Answer:  y = 4x + 12    (edited to reflect a SQUARE)

Step-by-step explanation:

Opposite Diagonal of a square means the line goes through the midpoint and has the opposite reciprocal slope.

Use the Midpoint formula:     $M=\bigg(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\bigg)$

$M=\bigg(\dfrac{2-6}{2},\dfrac{3+5}{2}\bigg)\\\\\\.\ =\bigg(\dfrac{-4}{2},\dfrac{8}{2}\bigg)\\\\\\.\ =(-2,4)$

Use the Slope formula:   $m=\dfrac{y_2-y_1}{x_2-x_1}$

$m=\dfrac{3-5}{2+6}\quad =\quad \dfrac{-2}{8}\quad =\quad -\dfrac{1}{4}$

Opposite Diagonal slope is opposite (change the sign) and reciprocal (flip the fraction)  → m⊥ = + 4

Now use the Point-Slope formula:   y - y₁ = m⊥(x - x₁)    where

• (x₁, y₁) is the midpoint (-2, 4)
• m = 4

y - 4 = 4(x + 2)

y - 4 = 4x + 8

   y = 4x + 12

Answer 2

Simple question

slope of on le diagonal,whose coordinates are given=

$\frac{y2 - y1}{x2 - x1} = \frac{5 - 3}{ - 6 - 2} = \frac{2}{ - 8} or \: \frac{ - 1}{4}$

now we know that diagonals of square are perpendicular to each other

and we also know that if there are two perpendicular lines have slope m1 and m2 respectively,then

$m1 = \frac{ - 1}{m2}$

so if one diagonal's slope is -1/4 then other diagonal's slope will be 4

now we know that all the equation of a line is in the form

y=mx+c

where m is slope

so simply put the value of m=4

y=4x+c...i)

we also know that the line of equationi) will pass through the midpoint coordinates of first diagonal

mid point of first diagonal ={(2-6)/2 ,(5+3)/2}=(-2,4)

put this value of x and y in i) equation

4=-8+c

c=12

hence putting the value of c in equation I) we get the equation of line as

y=4x+12

y-4x-12=0