Welcome to a lesson on constructing an isosceles triangle.
Let's first review the definition of an isosceles triangle.
An isosceles triangle is a triangle with at least
two congruent sides. So looking at this triangle here,
this should an isosceles triangle if two or more sides
are congruent. And it does follow from the
isosceles triangle theorem, that the angles
opposite the congruent sides would also be congruent.
So it's not part of the formal definition of an isosceles triangle.
But it is true that this angle would be congruent to this angle here.
Okay, let's go ahead and talk about how we can construct an isosceles triangle.
In order to perform the construction we need a straight edge and a compass.
Next we would start with the segment of a specified length as we
see here.
And then if we're given the freedom to construct any isosceles triangle that we
want,
the only restriction is that the radius of our compass
must be more than half the length of this given segment.
So if we're given a specific length for
the remaining two sides of an isosceles triangle,
you can mark that length with your compass. But let's just say we're not given any
restrictions. So the only restriction is
the radius of the compass must be greater than
half the length of this segment. So if we put the point of the compass here,
and then the pencil of the compass here,
which looks like its past the midpoint, we're going to swing an arc
above this segment. So it might look something like this.
Now without adjusting the compass and that's important, we don't want to change
the radius of our compass. We're going to take the point and put it on the
other end point,
and then swing another arc above this segment.
So the second arc might look something like this.
Now we're going to use our straight edge
and make a segment from this endpoint to the intersection of the arcs.
And then another segment from this endpoint to the intersection the arcs
and we'll have our isosceles triangle.
This would be one side,
and this would be the other.
So notice how the two congruent sides of this triangle
have a length that's the radius of two congruent circles.
So we're guaranteed that this side here is congruent
with this side here and therefore we have an isosceles triangle.
Let's go ahead and show that one more time
where we use a larger radius to swing the arcs.
So again we have a segment of a specified length. We'll put one endpoint of the
compass
on one endpoint of the segment. And then just make sure that it's open past
the midpoint
of the given segment. So we put the point here and the pencil here.
We would then swing arc above the segment,
that would look like this.
And then without adjusting the compass
we'll put the point on the other endpoint of the segment, and swing another
arc
above a given segment, which might look like this.
And now we'll just make segments
from the intersection of the two arcs to the two endpoints
of the segment. And that'll be our isosceles triangle.
There's one side,
and there's the other side of our isosceles triangle.
Again these two sides here would be congruent
and therefore, we have an isosceles triangle. I just want to show you one more thing.
Remember how the radius of the compass must be
greater than half the length of this segment. Here's an example of why.
If we put the point of our compass here and the pencil of our compass here,
notice how it would not be longer than half the length of this segment. So if we swung
an arc
above this segment, and then without adjusting the compass,
put the point on the other endpoint and then swing another arc above this segment.
Notice how the arcs would not intersect because the radius
of the circle is not greater than half the length of this segment.
I hope you found this video helpful.