Exercise 11 Page 467

Let's modify the given figure a bit, keeping in mind that the question asks for the length of a segment. We focus on the length measures given on the figure and ignore the angle measures.

• We know that the incenter of a triangle is equidistant from the sides, so let's mark that FH, DH, and GH are congruent.
• Let's also highlight the segment BD — the length of which we are asked to find — and shade the right triangle △ BDH with this segment as one of the legs.
• Finally, let's shade the right triangle △ AFH, where the length of two sides are given.

Let's concentrate on these shaded triangles.

We can find BD in three steps.

Finding FH.

We can use the Pythagorean Theorem in the green triangle to find FH.

Finding DH.

We already noticed that FH is congruent to DH, so their measure is the same. DH=FH=7

Finding BD.

The red right triangle HB=11 is given on the diagram, and we now know the length of DH. We can use the Pythagorean Theorem, this time in the red triangle, to find BD.

The length of BD is BD=sqrt(72), or approximately 8.5.