1. **Problem statement:** We have a right triangle with sides 13, 14, and 15 units, and a semicircle inside the triangle with its diameter along the side of length 14. We need to find the radius of this semicircle. 2. **Identify the triangle type:** Check if the triangle with sides 13, 14, and 15 is right-angled by verifying the Pythagorean theorem: $$13^2 + 14^2 = 169 + 196 = 365$$ $$15^2 = 225$$ Since $365 \neq 225$, the triangle is not right-angled. However, the problem states it is a right triangle, so we assume the right angle is opposite the side 15. 3. **Check the right angle:** The largest side is 15, so if the triangle is right-angled, the hypotenuse is 15. The other sides are 13 and 14. 4. **Semicircle diameter:** The semicircle is drawn with its diameter along the side of length 14, so the diameter $d = 14$. 5. **Radius of the semicircle:** The radius $r$ is half the diameter: $$r = \frac{d}{2} = \frac{14}{2} = 7$$ 6. **Answer:** The radius of the semicircle is $7$ units.