Video Transcript
Given that 𝐴𝐵𝐶 is an equilateral
triangle with side length four centimeters, use a ruler and a compass to draw the
triangle 𝐴𝐵𝐶 and bisect angle 𝐴, angle 𝐵, and angle 𝐶 by the bisectors ray
𝐴𝐷, ray 𝐵𝐷, and ray 𝐶𝐷 that intersect at point 𝐷. If ray 𝐶𝐷 intersects line segment
𝐴𝐵 at point 𝑀, use the ruler to measure the length of line segment 𝐷𝑀 to the
nearest decimal.
Let’s begin by constructing the
equilateral triangle described with side length four centimeters. We begin by sketching one side of
the triangle with length four centimeters using a ruler. We name this side 𝐶𝐵. We know that an equilateral
triangle has sides of equal length. So, we use a compass to trace a
circle centered at point 𝐶 with radius four centimeters. Then, we repeat this process with
the compass centered at point 𝐵. The intersection of these circles
of radius four centimeters is the third vertex of the equilateral triangle, point
𝐴. Using a ruler, we connect the
vertices to form the required triangle.
Next, we are asked to bisect each
interior angle of triangle 𝐴𝐵𝐶. To bisect angle 𝐴, we first use a
compass to trace a circle centered at 𝐴 that intersects the sides of the angle. Then, we do the same for angles 𝐵
and 𝐶. To finish the first step, we’ll
mark the intersection points as shown. In the second step, we trace two
circles of the same radius centered at each point of intersection found in the
previous step. So the circles intersect on the
interior of angle 𝐶𝐴𝐵. Then, we can use our ruler to draw
a line connecting this point to vertex 𝐴. This is the bisector of angle
𝐴.
Next, we can follow the same
process to bisect angle 𝐵. We use a compass to trace two
circles of the same radius centered on the points of intersection found in the first
step. Then, we mark the intersection of
these two circles and draw the angle bisector from vertex 𝐵 through this point of
intersection.
Finally, we’ll construct the angle
bisector of 𝐶 using the same tools and steps. The third angle bisector looks like
this. As we know, the angle bisectors of
a triangle will always be concurrent. In other words, they intersect at a
single point. In this case, the point of
concurrency is named 𝐷.
In summary, we have constructed ray
𝐴𝐷, the bisector of angle 𝐴; ray 𝐵𝐷, the bisector of angle 𝐵; and ray 𝐶𝐷,
the bisector of angle 𝐶. We are told that ray 𝐶𝐷
intersects side 𝐴𝐵 at point 𝑀.
Now that we are done drawing the
diagram according to the directions, we are ready to measure the length of line
segment 𝐷𝑀. If we measure this length with a
ruler, we get 𝐷𝑀 equal to approximately one and two-tenths centimeters. To the nearest decimal, this
measurement is 1.2 centimeters.
