Given GEH = 35^ and EG = 25.4 cm.

Question

ScanSolve AI Tutor · Accepted Answer

Here are the steps to calculate the length of $EF$ :

Calculate the lengths of $EH$ and $GH$ in the right-angled triangle $\triangle GEH$ .
Given $\angle GEH = 35^\circ$ and $EG = 25.4$ cm.
$EH = EG \cos(35^\circ) = 25.4 \cos(35^\circ)$ $GH = EG \sin(35^\circ) = 25.4 \sin(35^\circ)$
Calculate $\angle EGH$ .
In $\triangle GEH$ , the sum of angles is $180^\circ$ . Since $\angle GHE = 90^\circ$ :
$\angle EGH = 180^\circ - 90^\circ - 35^\circ = 55^\circ$
Calculate $\angle HGF$ .
Given $\angle EGF = 120^\circ$ . We know $\angle EGF = \angle EGH + \angle HGF$ .
$\angle HGF = \angle EGF - \angle EGH = 120^\circ - 55^\circ = 65^\circ$
Calculate the length of $HF$ in the right-angled triangle $\triangle GHF$ .
We use the value of $GH$ from Step 1 and $\angle HGF$ from Step 3.
$HF = GH \tan(\angle HGF) = (25.4 \sin(35^\circ)) \tan(65^\circ)$
Calculate the total length of $EF$ .
$EF = EH + HF$ $EF = 25.4 \cos(35^\circ) + (25.4 \sin(35^\circ)) \tan(65^\circ)$ $EF \approx 25.4(0.81915) + (25.4(0.57358)) (2.14451)$ $EF \approx 20.80641 + (14.57893) (2.14451)$ $EF \approx 20.80641 + 31.26850$ $EF \approx 52.07491 \text{ cm}$
Round the answer to three significant figures.
$EF \approx \boxed{\text{52.1 cm}}$

Answer