triangle fgh is an equilateral triangle with sides measuring 34\\sqrt{3} units. what is the height of the…

Explanation:

Step1: Recall equilateral - triangle property

In an equilateral triangle, the height divides the base into two equal parts. If the side length of the equilateral triangle is (a = 34\sqrt{3}) units, then the base of one of the right - triangles formed by the height is (\frac{a}{2}=17\sqrt{3}) units.

Step2: Apply the Pythagorean theorem

Let the side length of the equilateral triangle be (a) and the height be (h). In the right - triangle formed by the height, we have (a^{2}=h^{2}+\left(\frac{a}{2}\right)^{2}). Substituting (a = 34\sqrt{3}), we get ((34\sqrt{3})^{2}=h^{2}+(17\sqrt{3})^{2}). [h^{2}=(34\sqrt{3})^{2}-(17\sqrt{3})^{2}] [h^{2}=34^{2}\times3 - 17^{2}\times3] [h^{2}=3\times(34^{2}-17^{2})] Using the difference - of - squares formula (x^{2}-y^{2}=(x + y)(x - y)) where (x = 34) and (y = 17), we have (h^{2}=3\times(34 + 17)\times(34 - 17)) [h^{2}=3\times51\times17] [h^{2}=3\times867] [h^{2}=2601] [h = 51]

Answer:

51 units