What are the properties of the incenter of a triangle?
7
Let's modify the given figure a bit, while also keeping in mind that the question asks for the length of a segment.
We will focus on the length measures given on the figure and ignore the angle measures.
Let's highlight the segments where the length is known.
We also know that the incenter of a triangle is equidistant from the sides. Let's mark that FH, DH, and GH are congruent.
We are asked to find DH. Since congruent segments have the same length, it is enough to find the length of one of FH, DH, and GH.
Notice that we know the length of two sides, HA=25 and AF=24, of the right triangle △ AFH.
We already concluded that DH is congruent to FH, so this also means that we found the length of DH.
DH=7
Extra
Can we draw a scaled diagram?
Some of you may wonder how to draw a scaled version of the diagram. Let's try. You can follow these steps either on paper with ruler and compass or using a geometry software.
We can start with the right triangle △ AFH, since the length of two sides are given.
We now have H, the center of the incircle, and F, the point where this incircle touches side AC (although we do not yet have the position of C). Let's draw this incircle.
Since AH bisects angle ∠ CAB, we can draw the line of AB by copying angle ∠ FAC to the other side of AH.
The next step is to find the position of vertex B. However, there are two measurements given on the diagram related to B and the position of the already constructed points: HB=11 and m∠ HBD=30. Which one shall we use?
In fact, there is no right triangle △ HBD with the already constructed HD=7, HB=11, and m∠ HBD=30.
If you construct B such that HB=11, then m∠ HBD≈ 39.5.
If you construct B such that m∠ HBD=30, then HB=14.
For the moment, you can experiment with these constructions. You will learn how to calculate these values when you will learn trigonometry.
We need to stop our construction here. The measurements on the sketch in the book do not allow us to draw a scaled diagram. This also means that, if you followed an approach different from ours, you may got a different value for DH.
