Measurement of Objects Using Similar Triangles in The Plane

John Gabrielson                Chicago H. S. for Agri. Science
                               3807 W. 111th Street 
                               Chicago IL  60655
                               (312)535-2500

Objectives:

The students in the high school geometry class will apply the use of similar 
triangles to a sight tool in order to measure the distance between, or height
of large objects such as buildings or trees.  
 
Materials Needed:

(Dimensions used can be adjusted for own use)
1/2 inch plastic tubing cut into 2' length for sight tube
1" x 2" x 4' wood base and arm
29cm x 33cm cardboard squares
30cm string cord
Washer weight
Bolt with wing nut and locking washer 
Copy of centimeter ruler
Long tape measure

Strategy:

    Construction of this measuring tool involves fastening two 1" x 2" x 4' 
boards with bolt wing nut and locking washer placed together approximately one 
inch from the end of both boards.  This will form the one arm and the base of 
the tool.   
    The 30cm string cord with a washer weight is attached to upper left hand 
corner of the 29cm x 33cm card.  This card with weighted cord is attached over 
the head of the bolt of the measuring tool at the edge of the joined stick.  A 
sight tube is glued onto the arm above the attached card at the top corner edge 
of the joined stick.  This forms the top upper corner of the measuring tool.  
Leveling the measuring tool will set the position where the weighted cord 
crosses the bottom of the card in a position exactly even with the position of 
its attachment to the top corner of the card.  This will help determine where 
the zero position of the copy of the centimeter ruler should be glued. 
    An angle of elevation will be formed by looking through the sight tube.  
The top of an object observed will form the local point.  This is point b of the 
triangle.  A line can be imagined across from the eye of the observer to the 
object observed.  On this object point c can be named.  The eye of the observer 
forms point a.  A large triangle has been formed by the thought process of 
connecting the points from the eye of the observer, to the top of the object 
observed, to the eye level on the object, back to the observer.  
     A second triangle similar to the above triangle will be formed onto the 
cardboard card attached to the measuring devise.  When the measuring devise is 
elevated the cord will move across the centimeter ruler glued at the bottom of 
the card.  When the cord stops moving, it will provide the measurement of a 
side.  Label this crossing, point e.  Point d can be named at the top of the  
where the cord is attached.  Point f can be named at the bottom corner of the 
card where the card forms a right triangle.  Segment df will be the side of the 
card and the other side of the small triangle segment ef will be the measurement 
along the centimeter ruler. 
     In order to solve these similar triangles the observers will have to know 
either the height of the object to find the distance or the distance to the 
object to find the height.  One then can set up the equality:        

                bc  =  ac                                                                   .
                ef     df

Having three knowns, two on the card, and one from the triangle of the object 
observed, one can solve the problem. 
      The teacher will use this tool in a lesson to instruct students about 
similar triangles.  The students can be taught the concept of similar triangles 
using a proportional growth pattern of triangles.  This can be followed by 
instructional use of the tool.  The students are then to be brought outside and 
begin the actual computation of similar triangles by using the measuring tool.

Performance Assessment:

The students can be evaluated in groups.  Collection of group work will reveal 
how accurately the measurements were taken.  Correct mathematics can also be 
checked.  The scores given can then be recorded and individuals will be given a 
group grade.