More Triangle Problems, 7

Category: Plane Geometry

"Published in Vacaville, California, USA"

A pole 26 ft. long leans against a wall at a point 10 ft. from the ground. What is the length of the projection of the pole on the ground?

Solution:

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

Photo by Math Principles in Everyday Life

Let's assume that the wall and the ground are perpendicular to each other.

Let y = be the length of the projection of the pole on the ground. 

By Pythagorean Theorem, the length of the projection of the pole on the ground is

Similar Triangles, 2

Category: Plane Geometry

"Published in Vacaville, California, USA"

A street light is 15 ft. directly above the curb. A man 6 ft. tall starts 10 ft. down street from the light and walks directly across the street which is 10 ft. wide. When he reaches the opposite curb, what is the distance between the initial and final positions of the tip of his shadow?

Solution:

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

Photo by Math Principles in Everyday Life

Let x = be the length of his shadow at the 1st curb.

By using similar triangles

 

 

 

   
If a man walks 10 ft. directly across the street which is at the opposite curb, the figure above becomes

Photo by Math Principles in Everyday Life

In order to solve for the distance between the initial and final positions of the tip of his shadow, let's label further the above figure as follows

Photo by Math Principles in Everyday Life

Let a = be the distance of a man to the street light from 2nd curb
      b = be the length of his shadow at the 2nd curb
      y = be the distance between the initial and final positions of the tip of his shadow

By Pythagorean Theorem
 

 

 

 

By using similar triangles

 

 

Since the 1st Curb is parallel to the 2nd Curb, then the small and large triangles are similar to each other. If the small triangle is an isosceles right triangle, then the large triangle is also an isosceles right triangle.

By Pythagorean Theorem

 

which is the same as the length of a man's shadow at the 1st Curb.
  
Therefore, the distance between the initial and final positions of the tip of his shadow is
  

  

More Triangle Problems, 6

Category: Plane Geometry, Trigonometry

"Published in Newark, California, USA"

Show that, for the two triangles ABC and AXY (see figure),

Photo by Math Principles in Everyday Life

Solution:

Since the altitude of the two triangles are not given, then we can draw the altitudes from points B and X as follows

Photo by Math Principles in Everyday Life

Consider ∆ABC:

but

then the above equation becomes

Consider ∆AXY:

but

then the above equation becomes

Therefore,