Question
Find the area of hexagon ABCDEF in which $\text{BL}\perp\text{AD},\text{CM}\perp\text{AD},\text{EN}\perp\text{AD}$ such that AP = 6cm, PL = 2cm, LN = 8cm, NM = 2cm, MD = 3cm, FP = 8cm, EN = 12cm, BL = 8cm and CM = 6cm.
✓
Answer
In hexagon ABCDEF,
there are triangles and trapeziums,
AP = 6cm, PL = 2cm, LN = 8cm,
NM = 2cm, MD = 3cm, FP = 8cm,
EN = 12cm, BL = 8cm and CM = 6cm
PN = PL + LN = 2 + 8 = 10cm
ND = NM + MD = 2 + 3 = 5cm
LM = LN + NM = 8 + 2 = 10cm
AL = AP + PL = 6 + 2 = 8cm
Now are $\triangle ABL =\frac{ AL + BL }{2}$
$=\frac{8 \times 8}{2}=32 cm^2$
Area $\triangle CDM =\frac{ MD + CM }{2}$
$=\frac{3 \times 6}{2}=9 cm^2$
Area $\triangle DEN =\frac{ ND + EN }{2}$
$=\frac{5 \times 12}{2}=30 cm^2$
Area $\triangle APE =\frac{ AP + FP }{2}$
$=\frac{6 \times 8}{2}=24 cm^2$
Area trap. $BLMC =\frac{1}{2}( BL + CM ) \times LM$
$=\frac{1}{2}(8+6) \times 10$
$=\frac{1}{2} \times 14 \times 10=70 cm^2$
$\text { Area trap. PFEN }=\frac{1}{2}(FP+EN) \times PN$
$=\frac{1}{2} \times 20 \times 10=100 cm^2$
$\therefore$ Area of figure $ABCDEF =$ Area of $\triangle ABL +\triangle CDM +$ Area $\triangle DEN +$ Area $\triangle APF +$ Area trap. BLMC + Area trap. PFEN,
$=(32+9+30+24+70+100) cm^2$
$=265 cm^2$
there are triangles and trapeziums,
AP = 6cm, PL = 2cm, LN = 8cm,
NM = 2cm, MD = 3cm, FP = 8cm,
EN = 12cm, BL = 8cm and CM = 6cm
PN = PL + LN = 2 + 8 = 10cm
ND = NM + MD = 2 + 3 = 5cm
LM = LN + NM = 8 + 2 = 10cm
AL = AP + PL = 6 + 2 = 8cm
Now are $\triangle ABL =\frac{ AL + BL }{2}$
$=\frac{8 \times 8}{2}=32 cm^2$
Area $\triangle CDM =\frac{ MD + CM }{2}$
$=\frac{3 \times 6}{2}=9 cm^2$
Area $\triangle DEN =\frac{ ND + EN }{2}$
$=\frac{5 \times 12}{2}=30 cm^2$
Area $\triangle APE =\frac{ AP + FP }{2}$
$=\frac{6 \times 8}{2}=24 cm^2$
Area trap. $BLMC =\frac{1}{2}( BL + CM ) \times LM$
$=\frac{1}{2}(8+6) \times 10$
$=\frac{1}{2} \times 14 \times 10=70 cm^2$
$\text { Area trap. PFEN }=\frac{1}{2}(FP+EN) \times PN$
$=\frac{1}{2} \times 20 \times 10=100 cm^2$
$\therefore$ Area of figure $ABCDEF =$ Area of $\triangle ABL +\triangle CDM +$ Area $\triangle DEN +$ Area $\triangle APF +$ Area trap. BLMC + Area trap. PFEN,
$=(32+9+30+24+70+100) cm^2$
$=265 cm^2$
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