Figure 6. |
In the classroom:
<> For geometry or industrial arts classes to use in-class participation
have a team of two students construct a semicircle or a
full circle using the carpenter's square method. Have each team choose
one of the triangles in their construction formed by connecting push pins as shown in Figure 5
and the tracings along the legs of the carpenter's square. Measure the
lengths of the legs and the hypotenuse. Next use the Pythagorean theorem
to show the triangle is a right triangle or very close to a right
triangle.
<>
For a trigonometry class to use in-class participation have a team of two
students construct a semicircle or a full circle using the carpenter's
square method. Have each team choose one of the triangles in their
construction formed by connecting push pins as shown in Figure 5 and the
tracings along the legs of the carpenter's square. Using a protractor have
the students measure one of the angles formed by the hypotenuse and a leg
and measure the length of the hypotenuse. Using sine and cosine formulas
determine the lengths of the legs of the triangle. Next use the
Pythagorean theorem to show the triangle is a right triangle or very close
to a right triangle.
<> For a modeling class to
use in-class participation have a team of two students construct a
semicircle or a full circle using the carpenter's square method. Have each
team choose one of the triangles in their construction formed by
connecting push pins as shown in Figure 5 and the tracings along the legs
of the carpenter's square. Label one of the acute angles of the triangle
and
measure the length of the hypotenuse. Using sine and cosine formulas
determine expressions the lengths of the legs of the triangle. Next use
the Pythagorean theorem to show the triangle is a right triangle or very
close to a right triangle.
Animations and Interactive Excel
routine: To see an animations of the construction procedure
outlined above choose one of the following:
(For a free Quicktime movie player go
to
http://www.apple.com/quicktime/download/ )
To download or execute an interactive
Excel routine for generating the semicircle click here. In this routine you can choose the diameter. Choosing the
diameter between 1 and 12 generates a nice display.
To download or execute a Geometer's
Sketchpad animation for generating a full circle click
here. (Execution requires the Geometer's
Sketchpad software.)
Mathematical
Connections: Using the carpenter's square provides a technique for
drawing a circle that doesn't rely on a compass. We must prove that the dots at the intersection of the legs of
the carpenter's square are on the circumference of a circle.
Figure 7 will be helpful in order to
show that the dots lie on a circle. At vertex C is a dot with coordinates (x,y)
constructed using the carpenter's method. For convenience, we give the coordinates of the
push pin at A as (0, 0 ) and those at the push pin at B as (d, 0), where d
is the distance between the push pins.
Figure 7. |
<> In a geometry or industrial
arts class to prove that the dots recorded at the corner of the
carpenter's square lie on a circle we can use the diagram in Figure 7. We
proceed to find the equation of the circle containing the dots. Hint: use
the Pythagorean theorem to determine the legs of triangle ACB and apply it
again to the triangle ACB. Students may need an assist in developing the
requisite lengths. For more details click here.
<>
In a trigonometry class to prove that the dots recorded
at the corner of the carpenter's square lie on a circle we can use the
diagram in Figure 7. Label angle BAC as
and
determine the lengths of sides AC and BC in terms of trigonometric
functions of
and
the length of the hypotenuse. Note that the dashed line from vertex C is a perpendicular
to segment AB. This gives us two other right triangles. Use these triangles and
trigonometry to determine formulas for x and y. Finally show that (x, y)
lies on a circle by starting with x^{2} + y^{2},
substituting in the trigonometric expressions for x and y, and simplifying
to obtain the equation of a circle. Students may need an assist in
developing the trigonometric expressions for x and y. For more details
click here. An alternative proof using
similar triangles is also included.
<>
A project for a modeling class can
focus on developing a parametric expression
for the set of dots that are connected
to form the semicircle. The development in this case uses the same
approach as given for the trigonometry class. After it is shown that the
points (x,y) lie on a circle the derived parametric representation can be
used to plot the set of points. This is how the animations mentioned above
were developed. Click here for more details.