Concept explainers
To verify: The given points are the vertices of a square.
Explanation of Solution
Given information:
The four points
Formula used:
Distance formula between two points
A rhombus is a special kind of quadrilateral with all sides equal and square is a rhombus with equal lengths of diagonals.
Proof:
Consider the given four points
Plot the given points on coordinate plane and join the vertices to form the quadrilateral ABCD as,
From the above figure, calculate the distance between all the four points A, B, C and D using distance formula to measure the length of sides.
Recall that the distance formula between two points
So, lengths of sides AB, BC, CA and DA will be calculated as,
Recall that a rhombus is a special kind of quadrilateral with all sides equal and square is a rhombus with equal lengths of diagonals.
Since, all the sides are equal, so, the given points are of rhombus.
Now, calculate the lengths of diagonals AC and BD,
Since, the diagonals are also of equal lengths, so, this rhombus is a square.
Since, the quadrilateral formed from the given points has equal sides and equal diagonals, thus it is proved that the given points are the vertices of a square.
Chapter 1 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning