### Video Transcript

Which of the following is the
correct construction of two plus root 34 on a number line by using a compass and a
straightedge?

Let us start off by considering how
we can approach this construction, and we can compare this to the options at the end
of the video. For now, let us consider the
expression two plus root 34. Since the square root term will be
the most difficult part to represent on a number line, we should first figure out
how to do this. One way we can construct a square
root is to use the Pythagorean theorem. We recall that this relates the
sides of a right triangle. Specifically, we have that 𝑎
squared plus 𝑏 squared equals 𝑐 squared, where 𝑐 is the length of the hypotenuse
and 𝑎 and 𝑏 are the other two side lengths. Importantly, if we square root both
sides, then we have that the square root of 𝑎 squared plus 𝑏 squared is 𝑐, which
we can mark on the triangle.

Now, we can see that one way to
find root 34 is to find two square numbers, 𝑎 squared and 𝑏 squared, that sum to
34. This works if we choose 25 and
nine. Thus, root 34 equals the square
root of five squared plus three squared.

Let us see how we could represent
this on a number line. Using a straightedge or ruler, we
can mark numbers an even space apart on a line. For instance, one centimeter
between each number is a possibility. And we can do the same thing for a
vertical axis of numbers.

Now, let us draw a right triangle
with a base length of five units along the line and a height of three units up the
line, resulting in the following drawing. Then, using the Pythagorean
theorem, we can say that the hypotenuse of this triangle must be root 34. Therefore, we can use a compass
with the point at the origin and the pencil at the top-right vertex of the triangle,
which allows us to draw a semicircle of radius root 34, centered about the
origin. And we can thus see that where this
semicircle intersects the number line must be root 34.

Now, finally, we can find two plus
root 34. Since our number line has an even
space between each integer value, we know that two plus root 34 will be two such
spaces in the positive 𝑥-direction. For example, if we drew it with one
centimeter between each number, then we would need to measure two centimeters to the
right. Thus, two plus root 34 will be in
the position indicated. Comparing this to the given
options, we can see that our diagram corresponds to the fifth option. Hence, the correct construction of
two plus root 34 on a number line using a compass and a straightedge is option
(E).