Question

if fg = gh = 79 and eh = 50, what is ef? ef =

Explanation:

Step1: Apply the Pythagorean theorem in right - triangle EFG and EHG

Since FG = GH = 79 and EH = 50, and \(\triangle EFG\) and \(\triangle EHG\) are right - triangles with right - angle at E. In right - triangle EHG, by the Pythagorean theorem \(EG^{2}=GH^{2}-EH^{2}\). In right - triangle EFG, \(EF^{2}=FG^{2}-EG^{2}\). But since FG = GH, we can directly use the fact that in right - triangle EFG, \(EF^{2}=FG^{2}-EH^{2}\).

Step2: Substitute the given values

We know that FG = 79 and EH = 50. Substitute these values into the formula \(EF=\sqrt{FG^{2}-EH^{2}}\). So \(EF=\sqrt{79^{2}-50^{2}}=\sqrt{(79 + 50)(79 - 50)}\) (using the difference of squares formula \(a^{2}-b^{2}=(a + b)(a - b)\)). Then \(EF=\sqrt{129\times29}=\sqrt{3741}\approx61.16\).

Answer:

\(\sqrt{3741}\) (or approximately 61.16)