1. **Problem Statement:** Given two vertices of an equilateral triangle at points $(-5,3)$ and $(5,3)$, find the coordinates of the third vertex such that the origin $(0,0)$ lies inside the triangle. Given $\sqrt{3} = 1.7$. 2. **Formula and Important Rules:** - The length of each side of an equilateral triangle is the same. - The third vertex can be found by rotating the segment between the two given points by $\pm 60^\circ$ around one vertex. - The midpoint of the base is $M = \left(\frac{-5+5}{2}, \frac{3+3}{2}\right) = (0,3)$. - The length of the side $s$ is the distance between the two given points: $$s = \sqrt{(5 - (-5))^2 + (3 - 3)^2} = \sqrt{10^2 + 0} = 10$$ 3. **Finding the height of the equilateral triangle:** - Height $h = \frac{\sqrt{3}}{2} s = \frac{1.7}{2} \times 10 = 8.5$ 4. **Coordinates of the third vertex:** - The third vertex lies vertically above or below the midpoint by height $h$. - Since the base is horizontal, the third vertex coordinates are: $$\left(0, 3 \pm 8.5\right)$$ - So possible points are $(0, 11.5)$ or $(0, -5.5)$. 5. **Determining which vertex contains the origin inside the triangle:** - The origin $(0,0)$ lies inside the triangle only if the third vertex is below the base (since the base is at $y=3$). - So the third vertex is at $(0, -5.5)$. **Final answer:** The coordinates of the third vertex are $\boxed{(0, -5.5)}$.