Hint: Recall the characteristics
of isosceles triangles and how they might be related to properties of
circle and a straightedge
to construct two congruent segments with a common endpoint. Without
measuring, how can you guarantee these segments are congruent? From
these segments, construct an isosceles triangle.
- What are the characteristics
of an isosceles triangle? What properties of geometric figures do you
know that might be of help to you to construct an isosceles triangle
and guarantee that it is isosceles without using any measurement
- Now in a similar way, construct an isosceles triangle with Sketchpad.
|To construct a circle:
- Construct a segment to represent the radius of the circle.
- Select one of the segment’s endpoints, as the center, and the
- Choose Circle By Center + Radius under the Construct
construct a point on the circle:
- Select the
- Choose from
the Construct menu, the Point on Circle command.
Verify that the triangle you constructed is isosceles by
checking the lengths of the triangle’s sides.
Measure the triangle’s three interior angles, and classify your isosceles
Investigate the sum of the measures of the three interior
angles. Drag any vertex to
manipulate your triangle. Observe
what happens to the posted lengths and angle measurements.
What conclusions can you draw from your observations?
Discuss and prove a conjecture about angle measurement that
describes your findings.
Compare and contrast the effects of dragging each
of the triangle’s vertices around your sketch window.
How are the ways you may manipulate your triangle related to
your choice of vertex?
Discuss the differences between drawing and constructing a
Save your sketch onto a disk as isostri.gsp. Keep the
sketch open on your computer!
Sketchpad file illustrating the construction of an isosceles triangle has been saved as
What are the characteristics of an equilateral triangle?
Drag one of the vertices of your constructed isosceles
triangle until the triangle appears to be equilateral.
Without using measurement tools, how can you confirm that
this third side is congruent to the two radii of the circle?
an equilateral triangle that is
guaranteed to always be equilateral without the use of any measurement
tools. Hint: The Point
of Intersection command under the Construct
Click and drag
one of the vertices of your triangle.
How does it affect the equilateral triangle?
Verify that the triangle is equilateral by
measuring the length of the sides and the interior angles of the
drag one of the vertices. Observe
what happens to the posted lengths and angle measurements. Write, discuss
and prove a conjecture describing your findings.
are generalized recordings of sketches.
It is sometimes convenient to record the steps of particular
sketches (e.g., regular polygons) that you have made for subsequent playback.
For example, when a future sketch requires a constructed equilateral
triangle, you can just use your custom tool rather than reconstructing
one. Sketchpad can generate a
custom tool of a construction while you are sketching it.
can use custom tools repeatedly to generate figures, or portions of figures,
while sketching. Individual
custom tools can be used as foundations for larger sketches, leading to
potentially more complex constructions.
a Custom Tool:
Create a sketch of an example of the geometric
construction you want the tool to produce. You may use any
Sketchpad tools or menus to create this exemplar.
Select both the given objects (usually,
independent points) and the desired resulting objects you’d
like the tool to produce. The order in which you select the
givens determines the order in which you’ll match the givens
when using the tool.
Click and hold on the Custom tools
in the Toolbox. Choose Create New Tool from
the menu that appears.
You may name the tool in the dialog box that
appears, and click OK.
Your tool is added to the Custom Tools
menu, and is ready to use.
your Equilateral Triangle Custom Tool:
- Click and hold on the Custom tools
in the Toolbox. The Custom Tools menu appears.
- Choose your equilateral triangle tool from
- Move your mouse over the sketch and click in two different
places. An equilateral triangle appears in your sketch
- When you’re finished making an equilateral triangle, click
on any other tool in the Toolbox or press the Esc
- Euclid's first proposition in the Elements is, "For a line segment AB, there is an equilateral
triangle having the segment as one of its sides".
He provides a proof using a similar construction to the one you
have just completed. Provide your own proof that the triangle you constructed is
an equilateral triangle.
a custom tool for the construction of a right triangle.
Make sure to drag each vertex to confirm it stays a right triangle.
Be sure to save it for future
use on your disk. Describe
how you constructed your right triangle.
You may find the Perpendicular
Line command under the Construct
Sketchpad sketch illustrating the construction of a right triangle has
been saved as righttri.gsp.)