Chapter 12.8, Problem 26E

To determine

To find:The diagonals are perpendicular to each other without performing any calculations and the diagonals of any square are perpendicular to each other by using similar diagram.

Answer to Problem 26E

The diagonals are perpendicular to each other as they intersect at the origin and the diagonals of any square are perpendicular to each other as it can be expressed by using variable.

Explanation of Solution

Given information:

  

It is known that the coordinates of any square are perpendicular to each other if they intersect at the origin as the origin is the point at the axis $x,y$ intersect each other.

From the diagram it can be seen that the diagonals of the square $TUVW$ intersect at the origin.

Hence, the diagonals are perpendicular.

Now,

The similar diagram can be used to shown that the diagonals are perpendicular to each other. It can be generalized by using the variable instead of number. Thus,

The coordinates of $TUVW$ can be expressed as,

  $T=\left(0,a\right),U=\left(a,0\right),V\left(0,-a\right),W\left(-a,0\right)$

Therefore, the diagonals are perpendicular to each other as they intersect at the origin and the diagonals of any square are perpendicular to each other as it can be expressed by using variable.