MHB Area of Triangle ABC: Find the Answer Here

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The area of triangle ABC is confirmed to be 18. A participant discusses the relationship between the areas of triangles ACD and ABD, noting that ACD's area is three times that of ABD. Suggestions are made to extend line AC to point E, creating a similarity between triangles ACD and ECB. The conversation includes geometric coordinates and calculations to support the findings. Overall, the thread focuses on determining the area of triangle ABC through geometric relationships.
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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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