Quadrilateral Area 415A48

1. **State the problem:** We need to find the area of quadrilateral $EFGH$ with sides $EH=8$ cm, $HG=11$ cm, and $FG=4$ cm, and right angles at $H$ and $F$. 2. **Analyze the shape:** Since $H$ and $F$ are right angles, $EFGH$ can be split into two right triangles by the diagonal $EG$. 3. **Identify triangles:** Triangle $EHG$ is right-angled at $H$ with legs $EH=8$ cm and $HG=11$ cm. 4. **Calculate area of triangle $EHG$:** $$\text{Area}_{EHG} = \frac{1}{2} \times EH \times HG = \frac{1}{2} \times 8 \times 11 = 44 \text{ cm}^2$$ 5. **Calculate length of diagonal $EG$ using Pythagoras:** $$EG = \sqrt{EH^2 + HG^2} = \sqrt{8^2 + 11^2} = \sqrt{64 + 121} = \sqrt{185} \approx 13.6 \text{ cm}$$ 6. **Triangle $EFG$ is right-angled at $F$ with hypotenuse $EG=\sqrt{185}$ cm and side $FG=4$ cm.** 7. **Find length $EF$ using Pythagoras:** $$EF = \sqrt{EG^2 - FG^2} = \sqrt{185 - 16} = \sqrt{169} = 13 \text{ cm}$$ 8. **Calculate area of triangle $EFG$:** Since $F$ is a right angle, area is $$\text{Area}_{EFG} = \frac{1}{2} \times EF \times FG = \frac{1}{2} \times 13 \times 4 = 26 \text{ cm}^2$$ 9. **Calculate total area of quadrilateral $EFGH$:** $$\text{Area}_{EFGH} = \text{Area}_{EHG} + \text{Area}_{EFG} = 44 + 26 = 70 \text{ cm}^2$$ **Final answer:** The area of quadrilateral $EFGH$ is $70$ cm$^2$.