Question

BC is tangent to circle A at B and to circle D at C. What is AD to the nearest tenth?

Answer 1

Refer to the attached image. If we draw a line parallel to segment BC, and this line goes through point D, then we'll form the new point E. Point E is on segment AB.

The parallelogram EBCD parallelogram forms. In fact, this figure is actually a rectangle due to the right (90 degree) angles. By definition, the opposite sides are parallel. Consequently, the opposite sides are congruent.

So,
BC = ED = 24
EB = CD = 4

AE+EB = AB
AE+4 = 5
AE = 1

Note how triangle AED is a right triangle with the right angle at angle E.

We can use the pythagorean theorem to find x

a^2 + b^2 = c^2
1^2 + 24^2 = c^2
1 + 576 = c^2
577 = c^2
c^2 = 577
c = sqrt(577)
c = 24.0208

If we round to the nearest tenth, then we get 24.0 which is the final answer (so the answer is choice C)

Note: this is misleading as this implies that the hypotenuse is the same length as the leg, which is not the case. So this is one drawback to rounding.