Question

In the rectangle below, E1=2x -2, FH=3x+6, and m 4IHG =35°.Find Gl and m LIEH

Answer 1

Recall that the diagonals of a rectangle bisect each other and are congruent, therefore:

$\begin{gathered} FH=2EI, \\ GI=EI. \end{gathered}$

Substituting the given expression for each segment in the first equation, we get:

$3x+6=2(2x-2).$

Solving the above equation for x, we get:

$\begin{gathered} 3x+6=4x-4, \\ 3x+6+4=4x, \\ 3x+10=4x, \\ 4x-3x=10, \\ x=10. \end{gathered}$

Substituting x=10 in the equation for segment EI, we get:

$EI=2*10-2=20-2=18.$

Therefore:

$GI=18.$

Now, to determine the measure of angle IEH, we notice that:

$\Delta HFG\cong\Delta GEH,$

therefore,

$\measuredangle GHF\cong\measuredangle HGE.$

Using the facts that the triangles are right triangles and that the interior angles of a triangle add up to 180° we get:

$m\measuredangle IEH=90^{\circ}-35^{\circ}=55^{\circ}.$

Answer:

$\begin{gathered} m\operatorname{\measuredangle}IEH=55^{\operatorname{\circ}}, \\ GI=18. \\ \end{gathered}$