# The location of the point S so that quadrilateral PQRS is a square and area of the square along with figure.

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1.8, Problem 22E
To determine

## To calculate: The location of the point S so that quadrilateral PQRS is a square and area of the square along with figure.

Expert Solution

The coordinates of point S so that PQRS forms a square is S(0,4) and its area is 50 sq. units .

### Explanation of Solution

Given information:

The points P(5,1),Q(0,6) and R(5,1) .

Formula used:

A square is a special kind of quadrilateral having equal lengths of its sides and diagonals.

Distance formula between two points X(a1,b1) and Y(a2,b2) is calculated as,

d(XY)=(a2a1)2+(b2b1)2

Area of a square with its side length x is the square of its side, which is mathematically written as,

Area=(side)2=x2

Calculation:

Consider the provided vertices P(5,1),Q(0,6) and R(5,1) .

By plotting the given points on the coordinate plane, we get the following figure,

Recall that a square is a special kind of quadrilateral having equal lengths of its sides and diagonals.

So, to find the coordinates of S such that PQRS forms a square, the lengths of its diagonals i.e. PR and QS must be equal.

PR¯ = QS¯

For the coordinates to form a square, S must lie on y-axis, so, assume the coordinates of S as (0,y) .

Now, apply distance formula between two points X(a1,b1) and Y(a2,b2) which iss calculated as,

d(XY)=(a2a1)2+(b2b1)2

PR¯ = QS¯(5(5))2+(11)2=(00)2+(6y)2102=(6y)210=6y

Simplify it further as,

10=6yy=610y=4

Therefore, the fourth vertex of the square PQRS is at point S(0,4) .

Now, the length of side of square is calculated as,

PQ=(05)2+(61)2=52+52=252=52

Recall that area of a square with its side length x is the square of its side, which is mathematically written as,

Area=(side)2=x2

So, area of square PQRS is calculated as,

Area=(side)2=(52)2=252=50 sq. units

Thus, the coordinates of point S so that PQRS forms a square is S(0,4) and its area is 50 sq. units .

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