Trapezoid, Quadrilateral Problems, 2

Category: Plane Geometry

"Published in Vacaville, California, USA"

 A sail has a spread of canvass as shown in the figure. Find the surface area of one side of the sail.

Photo by Math Principles in Everyday Life

Solution:

The given sail canvass is a general quadrilateral because the two opposite sides are not parallel. Because of this, we need to divide this into two triangles as follows

Photo by Math Principles in Everyday Life

The first triangle is a right triangle because the two adjacent sides are perpendicular to each other. Consider the right triangle at the left side. The area of a first triangle is

Consider the general triangle at the right side. In order to get the value of x, use Pythagorean Theorem at the first triangle, we have
 

 

 

 

 

 

   
Since the other side of a second triangle is also 15 ft., then the second triangle is an isosceles triangle. Use Heron's Formula in order to get the area of the second triangle, we have
 

   
where a, b, and c are the sides of a triangle and s is the semi-perimeter of a triangle.

The perimeter of a triangle is
 

 

 

   
The semi-perimeter of a triangle is
 

 

 

   
Hence, the area of a second triangle is
 

 

 

 

   
Therefore, the area of a sail canvass is
 

 

Trapezoid, Quadrilateral Problems

Category: Plane Geometry

"Published in Vacaville, California, USA"

The vertical end of a trough has the following dimensions: width at the top 4.4 ft., width at the bottom 3.2 ft., depth 3.5 ft. Find the area of the end of the trough.

Solution:

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

Photo by Math Principles in Everyday Life

From the description of the given word problem, the vertical end of a trough is a trapezoid. The area of a trapezoid is given by the formula

Substitute the values of h, b₁ and b₂ to the above equation, we have

Square, Rectangle, Parallelogram Problems, 9

Category: Plane Geometry

"Published in Newark, California, USA"

A certain city block is in the form of a parallelogram. Two of its sides are each 421 ft. long; the other two sides are each 227 ft. in length. If the distance between the first pair of sides is 126 ft., find the area of the land in the block, and the length of the diagonals.

Solution:

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

Photo by Math Principles in Everyday Life

Since the height of a parallelogram is given as well as the opposite sides, then we can solve for the area as follows

In order to solve for the diagonals of a parallelogram, we need to label further the figure above as follows

Photo by Math Principles in Everyday Life

Use Pythagorean Theorem for ∆ADE, we have

Use Pythagorean Theorem for ∆DEB, we have

Photo by Math Principles in Everyday Life

Use Pythagorean Theorem for ∆ACF, we have