10.002. Find the diagonal and the lateral side of an isosceles trapezoid with bases 20 and 12 cm, given that the center of the circumscribed circle lies on the larger base of the trapezoid.
## Solution.

Since $A D$ is the diameter of the circle (Fig. 10.2), then $O D = O C = 10$ cm. Draw $C L \perp A D$; then $O L = 6$ cm and from $\triangle C L O$ we find

$$
C L = \sqrt{O C^{2} - O L^{2}} = 8 \, \text{cm}
$$

Then from $\triangle A L C$ and $\triangle C L D$ we have

$$
A C = \sqrt{C L^{2} + A L^{2}} = \sqrt{64 + 256} = 8 \sqrt{5} \, \text{cm}
$$

and

$$
C D = \sqrt{C L^{2} + L D^{2}} = \sqrt{64 + 16} = 4 \sqrt{5} \, \text{cm}.
$$

Answer: $4 \sqrt{5}$ and $8 \sqrt{5}$ cm.

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-557.jpg?height=372&width=404&top_left_y=94&top_left_x=124)

Fig. 10.3

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-557.jpg?height=308&width=600&top_left_y=156&top_left_x=636)

Fig. 10.4