Question

# In the adjoining figure D is a point on BC such that, ∠ABD = ∠CAD. If AB = 5cm, AD = 4cm and AC = 3cm. Find A (∆ACD) : A (∆BCA).

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Hint: Visualizing the given data in the question draws an appropriate figure. According to the properties of triangles, observe similarities between the two triangles in order to establish a relation between their angles or sides.

Given data,
∠ABD = ∠CAD. AB = 5cm, AD = 4cm and AC = 3cm.

Let us compare ∆ABC and ∆CAD
⟹ $\dfrac{{{\text{A}}\left( {\Delta {\text{ACD}}} \right)}}{{{\text{A}}\left( {\Delta {\text{BCA}}} \right)}} = {\text{ }}\dfrac{{{\text{A}}{{\text{D}}^2}}}{{{\text{A}}{{\text{B}}^2}}}$
⟹ $\dfrac{{{\text{A}}\left( {\Delta {\text{ACD}}} \right)}}{{{\text{A}}\left( {\Delta {\text{BCA}}} \right)}} = {\text{ }}\dfrac{{{4^2}}}{{{5^2}}}$
⟹ $\dfrac{{{\text{A}}\left( {\Delta {\text{ACD}}} \right)}}{{{\text{A}}\left( {\Delta {\text{BCA}}} \right)}} = {\text{ }}\dfrac{{16}}{{25}}$