The centers of two circles are E and G, and they have radii 5 cm and 12 cm respectively. The distance between E and G is 13 cm. Show that the angle at F is 90 degrees.

Step 1: Show that $\angle EFG = 90^\circ$.
Given the radii of the circles and the distance between their centers:

• Radius of circle E, $EF = 5 \text{ cm}$.
• Radius of circle G, $GF = 12 \text{ cm}$.
• Distance between centers, $EG = 13 \text{ cm}$.

Consider the triangle $\triangle EFG$. We check if the Pythagorean theorem holds for its sides:

$$EF^2 + GF^2 = 5^2 + 12^2 = 25 + 144 = 169$$ $$EG^2 = 13^2 = 169$$

Since $EF^2 +