To inscribe a circle in the triangle, we have to find it's incenter.
Practice makes perfect
Copying the triangle
To copy the triangle, we can use the ASA Congruence Theorem. By copying the horizontal side and the two angles that this side helps to creates, we can draw a congruent triangle. Measuring the horizontal side, we see that it's 4 cm. We will start by drawing the horizontal side along the x-axis.
Using a protractor, we can measure an angle of 135^(∘) at the vertex that we have labeled A.
We will repeat the procedure for the 20^(∘) angle at the vertex labeled B.
Now we can draw the triangles remaining two sides using a ruler.
Let's isolate the triangle.
To inscribe a circle, we have to find the triangle's incenter. We can do that by drawing angle bisectors for at least two of the triangle's vertices. The point where the angle bisectors intersect is the triangle's incenter.
Finding the incenter
Let's start by drawing the angle bisector for ∠ B. First, we will draw an arc across AB and BC with an arbitrary radius like below.
Next, we draw two smaller arcs using the intersections of the first arc with AB and BC as our centers. Make sure you keep the compass settings the same for both of them.
The segment from B and through the intersection point of the two smaller arcs is the angle bisector to ∠ B.
We need one more angle bisector to find the incenter. Let's repeat the procedure for ∠ A.
Where the angle bisectors intersect, we find the triangle's incenter. Let's isolate this in our diagram.
To inscribe a circle, we also need to find the perpendicular bisector from the incenter and one of the triangle's sides. Let's do that with BC.
Finding a perpendicular bisector
We have to draw an arc that intersects BC as well as two smaller arcs using the intersections of the first arc with BC as our centers. Make sure you keep the compass settings the same for the two smaller arcs.
The segment from D to the intersection of the two smaller arcs is the perpendicular bisector.
Let's limit the perpendicular bisector to DE.
Constructing the inscribed circle
Finally, we will place the compass at D and set it's width to DE. Now we can construct the inscribed circle.
