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

Circular Arc Problems

Category: Plane Geometry

"Published in Vacaville, California, USA"

Los Angeles and New York are 2450 mi apart. Find the angle that the arc between these two cities subtends at the center of the earth. (The radius of the earth is 3960 mi.)

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 distance of two places. If you know the position of a city or a place like latitude and longitude, then you can calculate the central angle of two cities or places using the principles in solving spherical triangles. After the calculation of central angle, the distance of two places can be calculated. Since the distance of two cities is given in the problem, then we can proceed in calculating the central angle of two cities as follows

where S is the length of circular arc, R is the radius of a circle, and θ is the central angle of 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 central angle as follows

You can also express the value of central angle in degrees as follows

or

 

Circular Sector Problems

Category: Plane Geometry

"Published in Vacaville, California, USA"

A sector in a circle of radius 25 ft has an area of 125 ft². Find the central angle of the sector.

Solution:

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

Photo by Math Principles in Everyday Life

The area of a circular sector is given by the formula

where A is the area of a circular sector, R is the radius of a circular sector, and θ is the central angle of circular sector in radians. Radians is a unit less value of an angle.

Substitute the values of A and R in order to solve for the value of central angle, as follows

You can also express the value of central angle in degrees, as follows

or

Regular Polygon Problems

Category: Plane Geometry

"Published in Vacaville, California, USA"

Find the perimeter of a regular hexagon that is inscribed in a circle of radius 8 m.

Solution:

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

Photo by Math Principles in Everyday Life

If you will draw the diagonals that passes through the center of a circle, there are six triangles in a regular hexagon as follows

Photo by Math Principles in Everyday Life

The center of a circle bisects the three diagonals of a regular hexagon. The bisected diagonals are equal to the radius of a circle which is 8 m. The vertex angle of each triangles can be calculated as follows

Photo by Math Principles in Everyday Life

The triangles in hexagon are isosceles triangles because the two sides of each triangles are congruent which is 8 m. If the two sides of an isosceles triangle are congruent, then its base angles are congruent also. We can calculate the base angle of an isosceles triangle as follows

Since the base angles of an isosceles triangle are the same as the vertex angle which is 60°, then the isosceles triangle is an equiangular or equilateral triangle. Hence, the sides of an hexagon is x = 8 m. 

Therefore, the perimeter of a regular hexagon is