Area of Triangle ABC: Find the Answer Here

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SUMMARY

The area of triangle ABC is definitively calculated to be 18 square units. The discussion highlights the relationship between the areas of triangles ACD and ABD, noting that the area of ACD is three times that of ABD. A suggestion is made to extend line AC to point E, establishing similarity between triangles ACD and ECB, which aids in understanding the area calculations. The geometric properties and relationships are crucial for solving the problem effectively.

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what is the area of triangle ABC in the attached? answer is 18

i can not construct any similar triangles here. all i can see is area of ACD is 3 times area of ABD but how does it help me...
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ketanco said:
what is the area of triangle ABC in the attached? answer is 18

i can not construct any similar triangles here. all i can see is area of ACD is 3 times area of ABD but how does it help me...

How about extending AC to E such that ACD is similar to ECB?

[TIKZ]
\def\x{sqrt(265)/4}
\def\gamma{atan2(3,16)}
\coordinate[label=above:A] (A) at ({4*\x - 12 * cos(\gamma)},{12 * sin(\gamma)});
\coordinate[label=left:B] (B) at (0,0);
\coordinate[label=right:C] (C) at ({4*\x},0);
\coordinate[label=below:D] (D) at ({\x},0);
\coordinate[label=above:E] (E) at ({4*\x - 16 * cos(\gamma)},{16 * sin(\gamma)});

\draw[rotate={270-\gamma}] (A) +(0.4,0) -- +(0.4,0.4) -- +(0,0.4);
\draw[rotate={270-\gamma}] (E) +(0.4,0) -- +(0.4,0.4) -- +(0,0.4);

\draw (C) -- node[above] {12} (A) -- node[above left] {5} (B);
\draw (A) -- (D);
\draw (A) -- (E) -- (B);
\path (B) -- node[below] {$x$} (D) -- node[below] {$3x$} (C);
\draw[blue, ultra thick] (A) -- (B) -- (C) -- cycle;
[/TIKZ]
 
Klaas van Aarsen said:
How about extending AC to E such that ACD is similar to ECB?

[TIKZ]
\def\x{sqrt(265)/4}
\def\gamma{atan2(3,16)}
\coordinate[label=above:A] (A) at ({4*\x - 12 * cos(\gamma)},{12 * sin(\gamma)});
\coordinate[label=left:B] (B) at (0,0);
\coordinate[label=right:C] (C) at ({4*\x},0);
\coordinate[label=below:D] (D) at ({\x},0);
\coordinate[label=above:E] (E) at ({4*\x - 16 * cos(\gamma)},{16 * sin(\gamma)});

\draw[rotate={270-\gamma}] (A) +(0.4,0) -- +(0.4,0.4) -- +(0,0.4);
\draw[rotate={270-\gamma}] (E) +(0.4,0) -- +(0.4,0.4) -- +(0,0.4);

\draw (C) -- node[above] {12} (A) -- node[above left] {5} (B);
\draw (A) -- (D);
\draw (A) -- (E) -- (B);
\path (B) -- node[below] {$x$} (D) -- node[below] {$3x$} (C);
\draw[blue, ultra thick] (A) -- (B) -- (C) -- cycle;
[/TIKZ]
i see... thanks !
 

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