Square, Rectangle, Parallelogram Problems, 3

Category: Analytic Geometry, Plane Geometry

"Published in Newark, California, USA"

Three vertices of a parallelogram are (1, 3), (0, 0), and (4, 0). Find the three possible locations of the fourth vertex.

Solution:

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

Photo by Math Principles in Everyday Life

There are three possible locations of the fourth vertex of a parallelogram. The two points or vertices can be used to draw one of the sides or one of the diagonals of a parallelogram.

Case 1: If you will use the three points or vertices to draw the two sides of a parallelogram, the fourth vertex is located at the first quadrant as shown in the figure

Photo by Math Principles in Everyday Life

Since one of the sides of a parallelogram is a horizontal line that lies along the x-axis, the fourth vertex of a parallelogram is P(1 + 4, 3) or P (5, 3).

Case 2: If you will use the three points or vertices to draw the two sides of a parallelogram, the fourth vertex is located at the second quadrant as shown in the figure

Photo by Math Principles in Everyday Life

Since one of the sides of a parallelogram is a horizontal line that lies along the x-axis, the fourth vertex of a parallelogram is P(1 - 4, 3) or P (-3, 3).

Case 3: If you will use the three points or vertices to draw the two diagonals of a parallelogram, the fourth vertex is located at the fourth quadrant as shown in the figure

Photo by Math Principles in Everyday Life

The center of a parallelogram which is the intersection of two diagonals bisects the diagonals into two equal parts. In this case, the midpoint of a diagonal that lies along the x-axis is (2, 0). If the midpoint of the other diagonal is also (2, 0), then we can solve for the coordinates of the other end of other diagonal which the fourth vertex of a parallelogram as follows

Therefore, the coordinates of the other end of other diagonal or the fourth vertex of a parallelogram is P(3, -3).

Regular Polygon Problems, 2

Category: Analytic Geometry, Plane Geometry

"Published in Newark, California, USA"

A regular hexagon of side 6 has its center at the origin and one diagonal along the x-axis. Find the coordinates of its vertices.

Solution:

A regular hexagon has 3 longest diagonals that passes thru the center. There are 9 diagonals of a regular hexagon in total. In this problem, let's consider a longest diagonal that lies along the x-axis. To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

There are 6 triangles inside the regular hexagon. Since all sides of a regular hexagon are equal, then it follows that the three longest diagonals are equal to each other and bisect each other at the center. If the two sides of each triangles are equal, then all triangles are isosceles triangles. Let's further analyze and label the figure as follows

Photo by Math Principles in Everyday Life

The vertex angle of each triangles can be calculated as follows
 

 

If the two sides of an isosceles triangle are congruent, then it follows that the base angles are congruent also. The base angle of an isosceles triangle is

Since all angles of a triangle are congruent, then all triangles in a regular hexagon are equiangular or equilateral. Let's consider one triangle in a regular hexagon in order to calculate the altitude or height, we have

Photo by Math Principles in Everyday Life

Apply Pythagorean Theorem in order to solve for the altitude or height, we have
 

 

 

 

 

 

   
Therefore, the coordinates of the vertices of a regular hexagon are

Square, Rectangle, Parallelogram Problems, 2

Category: Analytic Geometry, Plane Geometry

"Published in Newark, California, USA"

The diagonals of a square of side 4 lie on the axes and its center at the origin. Find the coordinates of its vertices.

Solution:

If the center of a square is located at the origin, then the diagonals will be bisected into equal parts. Since the diagonals are located along the axes, then the sides of a square will be the hypotenuse of the four isosceles right triangles. To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

Consider the right triangle at the first quadrant and apply Pythagorean Theorem, we have 

Therefore, the coordinate of the vertices are

 

Square, Rectangle, Parallelogram Problems

Category: Analytic Geometry, Plane Geometry

"Published in Newark, California, USA"

A square of side 4 has its center at the origin and sides parallel to the axes. Find the coordinates of its vertices.

Solution:

If the center of a square is located at the origin, then the diagonals will be bisected into equal parts. Also, if the center of a square is located at the origin, then the axes will bisect the sides of a square into equal parts. To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

Therefore, the coordinates of the vertices are

V₁ (2, 2), V₂ (2, -2), V₃ (-2, -2) and V₄ (-2, 2).

Circular Arc Problems, 2

Category: Plane Geometry, Physics

"Published in Vacaville, California, USA"

How many revolutions will a car wheel of diameter 28 in. make over a period of half an hour if the car is traveling at 60 mi/hr?

Solution:

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

Photo by Math Principles in Everyday Life

This is a good application of circular arc problems in getting the number of revolutions of a car wheel especially when you're driving a car. In this problem, the distance traveled is not given but the speed or velocity and time are given. If you know the speed and time,then you can calculate the distance traveled as follows

where S is the distance traveled, V is the speed or velocity, and t is the travel time. Substitute the values of V and t, we have

Since the radius of a car wheel is expressed in inches, then we have to convert the distance traveled by car in inches as follows

Finally, we can get the number of revolutions of a car wheel as follows

where S is the total distance traveled or total length of a circular arc, R is the radius of a circle, and θ is the total angle of a circular arc in radians. Radians is a unit less value of an angle. 

Substitute the values of S and R in order to solve for the value of total angle, we have

Therefore, the number of revolutions of a car wheel is