question_answer

In the given fig. what is the ratio of the areas of

(a) shaded portion I to shaded portion II?

(b) shaded portion II to shaded portion III?

(c) shaded portions I and II taken together and shaded portion III?

Answer:

By splitting the above figure, we get,

(a) Now,    \[~AD=5\]

\[AB=AEBE=105\]

\[AB=5\]

Area of shaded portion I = Area of ABCD

\[\therefore \]  Area of \[ABCD=AD\times AB=5\times 5=25\]

So, Area of shaded portion \[I=25\]

Area of shaded portion II = Area of DCIJ ? Area FHI

Now,                 \[DJ=AJAD=105=5\]

\[DC=AB\](as It?s a square)

\[DC=5\]

of                     \[DCIJ=DJ\times DC=5\times 5=25\]

Now,                 \[AJ=EH\]    (side of a square)

\[EH=10\]

So,                   \[FH=EHEF=107\]

\[FH=3\]

and                   \[GF=BE\]

\[GF=5\]

area                  \[GFHI=GF\times FH\]

\[=5\times 3=15\]

Area o shaded portion \[II=25+15=40\]

So, required ratio of shaded portion I to II is \[25:40\]

\[=\frac{25}{40}=\frac{5\times 5}{8\times 5}\]

\[=\frac{5}{8}=5:8\]

(b) Area of shaded portion III = Area of BEFG

\[=BE\times EF\]

\[=5\times 7\]

\[=35\]

So, required ratio of shaded portion II to III is \[40:35\]

\[=\frac{40}{35}=\frac{8\times 5}{7\times 5}=\frac{8}{7}\]

\[=8:7\]

(c) Area of shaded portions I and II = 25 + 40

= 65

So, required ratio \[=65:35\]

\[=\frac{65}{35}=\frac{13\times 5}{7\times 5}=\frac{13}{7}\]

\[=13:7\]