In the figure, what is the length
of line segment 𝐸𝐺?
In the diagram, we can observe that
we have the triangle 𝐷𝐸𝐺, which has the line 𝐷𝐹 passing through it. Given that we have two line
segments that are both given as 10.5 length units, then we know that these sides are
congruent. And by definition, that means that
triangle 𝐷𝐸𝐺 is an isosceles triangle.
Now, let’s consider the line
𝐷𝐹. From the markings on the diagram,
we notice that line 𝐷𝐹, which passes through the vertex 𝐷, meets the base of line
segment 𝐸𝐺 at 90 degrees. And we can recall an important
theorem regarding these properties.
It is that the straight line that
passes through the vertex angle of an isosceles triangle and is perpendicular to the
base bisects the base and the vertex angle. So, because the angle at the vertex
is bisected, then the measure of angle 𝐺𝐷𝐹 equals the measure of angle
And importantly for this question,
we know that the base is bisected. So the line segment 𝐸𝐹 has the
same length as line segment 𝐺𝐹, which is 6.5 length units.
We are asked to find the length of
the line segment 𝐸𝐺, which we can find by adding the two lengths on the base. And 6.5 plus 6.5 gives us the
answer that the length of line segment 𝐸𝐺 is 13, or 13 length units.