Answer:
a) CD = 9
b) AB = 20
Step-by-step explanation:
<h3>a)</h3>
In this geometry, all of the right triangles are similar. This means The ratio of short side to long side is the same for all of the triangles.
You are given the short and long sides of ΔADB, and the long side of ΔCDA. You are asked for the short side of ΔCDA, so you can write the proportion ...
CD/AD = AD/BD
CD/12 = 12/16
CD = 12(12/16)
CD = 9
__
<h3>b)</h3>
There are a couple of options for finding AD. One you may be familiar with is the Pythagorean theorem.
AB² = AD² +DB²
AB² = 12² +16² = 144 +256 = 400 . . . . fill in known values
AB = √400 = 20 . . . . . take the square root
__
Alternatively, you can use the same proportional relationship that is described above. Here, we make use of the ratio of the hypotenuse to the long side.
AB/BD = CB/AB
AB² = BD·CB = 16·(16+9) = 16·25 . . . . cross multiply; fill in known values
AB = √(16·25) = 4·5 . . . . . take the square root
AB = 20
_____
<em>Additional comment</em>
This geometry, where the altitude of a right triangle is drawn, has some interesting properties. We have hinted at them above.
You can write three sets of proportions for this geometry: the ratios of short side and long side; the ratios of short side and hypotenuse; and the ratios of long side and hypotenuse. When you look at the way the sides touching the longest hypotenuse relate to that hypotenuse, you see three similar relations:
AC = √(CD·CB)
AD = √(DC·DB)
AB = √(BD·BC) . . . . . . . . the relation used in part (b) above
This "square root of a product" is called the <em>geometric mean</em>. In effect, the length of a side touching the longest hypotenuse is the geometric mean of the two segments of that hypotenuse that it touches.
