Start by drawing the shortest possible segment connecting C with AB.
2.4 units
Practice makes perfect
Let's look at the given triangle.
We can start by drawing the shortest possible segment connecting C with AB.
A diameter is the longest distance between the points on the circle. The length of the diameter determines the size of the circle. Therefore, to draw the smallest possible circle through C tangent to AB, we should use the shortest possible diameter, which is CD.
The circle intersects AC at J and BC at K.
Let's connect these points of intersection with the segment.
If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.
By this theorem we can see that JK is a diameter of the circle. Since CD is a diameter of the same circle, their lengths must be equal. Thus, to find JK we can calculate CD. To do so let's recall the Right Triangle Similarity Theorem, Theorem 9.6.
Right Triangle Similarity Theorem
If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.
In our case, it means that â–łACD ~ â–łCBD, â–łACD ~ â–łABC, and â–łABC ~ â–łCBD.
Knowing this, we can write a proportion relating CD with segments of the given length.
CD/AC=BC/AB ⇒
CD/3=4/5
Using this equality we can calculate CD.
