Triangle Inscribed in a Circle Problems, 2

Category: Plane Geometry

"Published in Newark, California, USA"

Find the area of the shaded region:

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Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

We noticed that the longest side of a triangle is also the diameter of a circle. If that's the case, the inscribed triangle is a right triangle. That is also a theorem. Let's prove that the triangle is a right triangle by Pythagorean Theorem as follows

Since both sides of the equation are equal, then the triangle is a right triangle. The base of a triangle is 12 and its altitude is 5.

Therefore, the area of the shaded region is

                              or

 

Square, Rectangle, and Parallelogram Problems, 16

Category: Plane Geometry

"Published in Vacaville, California, USA"

Al paints with watercolor on a sheet of paper 20 in wide by 15 in high. He then places this sheet on a mat so that a uniformly wide strip of the mat shows all around the picture. The perimeter of the mat is 102 in. How wide is the strip of the mat showing around the picture.

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

The first thing that we need to do is to get the length and width of the mat. 

The length of the mat is  and the width of the mat is . 

Hence, the perimeter of the mat is
 

 

 

 

 

 

Therefore, the width of the strip of the mat around the picture is 4 in. 

Circle and Rectangle Problems, 3

Category: Plane Geometry

"Published in Vacaville, California, USA"

A Norman window has the shape of a rectangle surmounted by a semicircle, as shown in the figure. If the perimeter of the window is 30 ft, express the area A of the window as a function of the width x of the window.

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

The given figure consists of a semicircle and a rectangle in which the diameter of a semicircle is equal to the width of a window. Let's label further the figure above as follows

Photo by Math Principles in Everyday Life

If the perimeter of a window is given in the problem, then the working equation is
 

 

 

 

 

 

 

 

Therefore, the area of a window as a function of x is
 

 

 

Circle and Rectangle Problems, 2

Category: Plane Geometry

"Published in Vacaville, California, USA"

A running track has the shape shown in the figure, with straight sides and semicircular ends. If the length of the track is 440 yd and the two straight parts are each 110 yd long, what is the radius of the semicircular parts (to the nearest yard)?

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above 

Photo by Math Principles in Everyday Life

Since the middle part of the running track is a rectangle, then the diameters of the two semicircular parts are equal. The width of a rectangle is equal to the diameter of two semicircles. If we combine the length of two semicircles, then it becomes a length or circumference of a circle. Hence, the length of a running track is

but  

Therefore, the radii of the semicircular parts of a running track are