Questions: C D is tangent to circle O at D. Find the diameter of the circle for B C=17 and D C=24. Round to the nearest tenth. (The diagram is not drawn to scale.)

C D is tangent to circle O at D. Find the diameter of the circle for B C=17 and D C=24. Round to the nearest tenth. (The diagram is not drawn to scale.)

Transcript text: 39. $\overline{C D}$ is tangent to circle $O$ at $D$. Find the diameter of the circle for $B C=17$ and $D C=24$. Round to the nearest tenth. (The diagram is not drawn to scale.)

Solution

Solution Steps

Step 1: Identify the Given Information

We are given:

$ BC = 17 $
$ DC = 24 $
$ CD $ is tangent to the circle at point $ D $

Step 2: Use the Pythagorean Theorem

Since $ CD $ is tangent to the circle at $ D $, $ OD $ is perpendicular to $ CD $. This forms a right triangle $ OBD $ with:

$ OB $ as the radius $ r $
$ BD $ as the hypotenuse
$ OD $ as one leg of the triangle

Step 3: Calculate $ BD $

Using the Pythagorean theorem in triangle $ BCD $: \[ BD^2 = BC^2 + CD^2 \] \[ BD^2 = 17^2 + 24^2 \] \[ BD^2 = 289 + 576 \] \[ BD^2 = 865 \] \[ BD = \sqrt{865} \] \[ BD \approx 29.4 \]

Step 4: Calculate the Radius $ r $

Since $ BD $ is the hypotenuse of the right triangle $ OBD $: \[ BD^2 = OB^2 + OD^2 \] \[ 29.4^2 = r^2 + r^2 \] \[ 865 = 2r^2 \] \[ r^2 = 432.5 \] \[ r = \sqrt{432.5} \] \[ r \approx 20.8 \]

Step 5: Calculate the Diameter

The diameter $ D $ of the circle is twice the radius: \[ D = 2r \] \[ D = 2 \times 20.8 \] \[ D \approx 41.6 \]