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Episode 24 – Ratio and Proportion and Right Triangles

In this episode I talk about a right triangle again and present the concept of scaling.  I also talk about an important safety subject on some youtube videos I watched last night.

The first picture shows the naming of the sides and angles in a right triangle.  The three angles in any triangle must add up to be 180 degrees.   So:.

Angle A + Angle B + 90 degrees = 180 degrees.   This means 90 degrees – Angle A = Angle B.    So we only have to determine one angle and all of the angles are known.

Angle A in a right triangle can be described by degrees or by Rise/ Run .
Rise is the length of the vertical line (Opposite side in this picture), and Run is the length of the horizontal line (Adjacent side in this picture).

 

The next picture shows how scaling works.   The large triangle is 4 X the size of the small triangle.   It is scaled by a ratio of 4:1.

If both triangles have the same angles, and only one side of the “unknown” triangle is known, then the scaling factor, or ratio can be determined.   Once that is known, all of the other sides can be easily determined.   In this next picture I know all the dimensions of the small triangle but I only know the length of the hypotenuse of the large triangle. Dividing the large hypotenuse byt the small hypotenuse gives me 15/5 = a scaling factor of 3, or a ratio of 3:1.

Now I can determine the opposite side length is the scale factor X the small triangle opposite side or 3 X 3 = 9.   Similarly, the adjacent side length is 3 X 4 = 12.

The above method can be used to find the height of a tree if the ground near the tree is level.    On a bright sunny day when the shadows are well defined, a pole is place in the ground.  The pole is checked to make sure it is plumb and the height of the pole is measured.  By simply measuring the length of the shadow of the pole and the length of the shadow of the tree we can determine the height of the tree by doing exactly the same calculations we did for the large triangle in the last picture.

Tree shadow / pole shadow = scaling factor.
scaling factor X pole height = tree height.
This works because the sun’s rays on a bright day are creating the same angle and both the pole and the tree form a right angle to the earth.

 

The next picture shows a device for drawing or measuring angles, a protractor.  The hole at the bottom is placed at the start of the angle where the two lines join together. The line on the protractor from the hole  is place over one of the lines forming the angle and the other line is measured on the scale.

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