1. **Problem statement:** Two trees stand on flat ground. The smaller tree is 7 m tall. The distance from the top of the smaller tree to the base of the taller tree is 15 m. The distance from the top of the taller tree to the base of the smaller tree is 20 m. We need to find: a) The horizontal distance between the two trees. b) The height of the taller tree. 2. **Formulas and rules:** We can model the problem using right triangles and the Pythagorean theorem. Let: - $x$ = horizontal distance between the two trees (unknown). - $h$ = height of the taller tree (unknown). The smaller tree height is 7 m. From the problem, the distances given form two right triangles: - Triangle 1: hypotenuse = 15 m, one leg = $x$, other leg = $h - 7$ (difference in heights). - Triangle 2: hypotenuse = 20 m, one leg = $x$, other leg = $h$ (height of taller tree). 3. **Set up equations using Pythagoras' theorem:** From triangle 1: $$15^2 = x^2 + (h - 7)^2$$ From triangle 2: $$20^2 = x^2 + h^2$$ 4. **Write the equations explicitly:** $$225 = x^2 + (h - 7)^2 = x^2 + h^2 - 14h + 49$$ $$400 = x^2 + h^2$$ 5. **Subtract the first equation from the second to eliminate $x^2 + h^2$:** $$400 - 225 = (x^2 + h^2) - (x^2 + h^2 - 14h + 49)$$ $$175 = 14h - 49$$ 6. **Solve for $h$:** $$14h = 175 + 49 = 224$$ $$h = \frac{224}{14} = 16$$ So, the taller tree is 16 m tall. 7. **Find $x$ using $h=16$ in the second equation:** $$400 = x^2 + 16^2 = x^2 + 256$$ $$x^2 = 400 - 256 = 144$$ $$x = \sqrt{144} = 12$$ 8. **Final answers:** a) The horizontal distance between the two trees is $12$ m. b) The height of the taller tree is $16$ m.