Question

find eg.
f
7
h
6w+5
e 5 i 4w+8 g
© algebra einstein, 2023

Explanation:

Step1: Use the Mid - segment Theorem

Since \(EH\) is parallel to \(EI\) (by the arrow marks indicating parallelism), by the Mid - segment Theorem of a triangle (a line segment joining the mid - points of two sides of a triangle is parallel to the third side and half its length), we have \(\frac{FI}{IH}=\frac{EI}{IG}\). Here, \(FI = 7\), \(IH\) is not needed as we can use the proportion of the divided sides. The theorem also gives \(\frac{EI}{IG}=\frac{FH}{HG}\). But more straightforwardly, since the line is a mid - segment (implied by parallelism and the way the sides are divided), we can set up the proportion \(\frac{EI}{IG}=\frac{FH}{HG}\). Wait, no, actually, if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. So \(\frac{EI}{IG}=\frac{FH}{HG}\). But since \(EH\) is parallel to \(EI\) (wrong, should be \(IH\) parallel to \(EF\)) no, wait, correct theorem: If a line is parallel to one side of a triangle and intersects the other two sides, then \(\frac{EI}{EG}=\frac{FH}{FG}\). Wait, no, the basic proportionality theorem (Thales' theorem): If a line is parallel to one side of a triangle and intersects the other two sides, then \(\frac{EI}{IG}=\frac{FH}{HG}\). But actually, since \(IH\parallel EF\), we have \(\frac{EI}{IG}=\frac{FH}{HG}\). Substituting the values: \(\frac{5}{4w + 8}=\frac{7}{6w+5}\).

Step2: Cross - multiply

Cross - multiplying the proportion \(\frac{5}{4w + 8}=\frac{7}{6w+5}\) gives \(5(6w + 5)=7(4w + 8)\).
Expand both sides: \(30w+25 = 28w + 56\).

Step3: Solve for \(w\)

Subtract \(28w\) from both sides: \(30w-28w+25=28w - 28w+56\), so \(2w+25 = 56\).
Subtract 25 from both sides: \(2w=56 - 25\), so \(2w=31\), then \(w=\frac{31}{2}=15.5\).

Step4: Find \(EG\)

\(EG=EI + IG=5+(4w + 8)\). Substitute \(w = 15.5\) into \(4w+8\): \(4\times15.5+8=62 + 8=70\). Then \(EG=5 + 70=75\).

Answer:

\(EG = 75\)