Square, Rectangle, Parallelogram Problems, 10

Category: Analytic Geometry, Plane Geometry

"Published in Vacaville, California, USA"

Three vertices of a rectangle are (2, 5), (-3, 5), and (2, -1). Find the coordinates of the fourth vertex.

Solution:

The first thing that we need to do is to plot the three points as follows

Photo by Math Principles in Everyday Life

Method 1:

Did you notice that the opposite sides of a rectangle are parallel to x-axis and y-axis? AD is parallel to x-axis and BC and CD is parallel to y-axis and AB. Points A and D have the same value of y which is 5. If AD is parallel to BC, then the value of y at point C must be -1 also.

Points A and B have the same value of x which is 2. If AB is parallel to CD, then the value of x at point C must be -3 also. Therefore, the fourth vertex of a rectangle is C (-3, -1).  

Method 2: 

This method is easier because you will need to use the midpoint formula only. We know that the center of a rectangle is the intersection of its diagonals. Their diagonals are equal and bisect each other at its center. 

In this case, we can get the midpoint of a rectangle at point O by using the midpoint formula for BD as follows

Hence, the coordinates of the center of a rectangle is O (-½, 2).

If the midpoint of AC is also point O, then we can solve for the coordinates of point C by using the midpoint formula of AC as follows

Therefore, the fourth vertex of a rectangle is C (-3, -1).
  

Solving Sudoku Puzzle

Category: Arithmetic

"Published in Newark, California, USA"

Solve for the missing digits in the Sudoku puzzle:

Photo by Math Principles in Everyday Life

Solution:

Sudoku is a well-known number puzzle worldwide from easy to difficult level. It is started in 1986 by the Japanese puzzle company, Nikoli, under the name, Sudoku, meaning "single number". 

Until now, Sudoku is still a popular number puzzle worldwide that you will see mostly in newspapers and magazines. You can assign the digits from 1 to 9. The repetition of the digits are not allowed by row, column, and even in 3 x 3 square. You must be patient in assigning the digits because many people can solve the puzzle within 30 minutes to an hour or so depending with the hardness of the puzzle. 

If you want to learn how to solve the Sudoku puzzle especially the beginners, please watch this video and this is the beginning or easy level puzzle.

              
This video is courtesy by YouTube. Thanks to blueclickers who shared this wonderful video at YouTube.

Rectangular Parallelepiped Problem, 14

Category: Solid Geometry, Plane Geometry, Trigonometry

"Published in Newark, California, USA"

Find the angles that the diagonal of a rectangular parallelepiped 2 in. by 3 in. by 4 in. makes with the faces.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

Next, we need to draw a diagonal line at the top and bottom of the rectangular parallelepiped as well as its diagonal and angles as follows

Photo by Math Principles in Everyday Life

By using Pythagorean Theorem, the length of x is

By using Pythagorean Theorem, the length of d is 

Therefore, the value of an angle is

or

Next, we need to draw a diagonal line at the left and right of the rectangular parallelepiped as well as its diagonal and angles as follows

Photo by Math Principles in Everyday Life

By using Pythagorean Theorem, the length of y is 

Therefore, the value of an angle is

or

Next, we need to draw a diagonal line at the front and back of the rectangular parallelepiped as well as its diagonal and angles as follows

Photo by Math Principles in Everyday Life

By using Pythagorean Theorem, the length of z is 

 

Therefore, the value of an angle is 

or