# The altitude AD of $\Delta ABC$ in which $\angle A$ is obtuse and AD = 10cm. If BD = 10 cm and $CD=10\sqrt{3}cm$, determine $\angle A$.

Last updated date: 21st Mar 2023

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Answer

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Hint: Calculate the value of $\angle CAD\ and\ \angle BAD$using trigonometric ratios like sine, cosine and tangent.

Complete step by step answer:

In $\Delta ACD$, since $\Delta ACD$ is a right angled triangle,

$\begin{align}

& \tan \left( \angle CAD \right)=\dfrac{CD}{AD} \\

& =\dfrac{10\sqrt{3}}{10} \\

& =\sqrt{3} \\

& \Rightarrow \angle CAD={{\tan }^{-1}}\sqrt{3} \\

& \Rightarrow \angle CAD=\dfrac{\pi }{3} \\

\end{align}$

We will now find $\angle BAD$, since $\Delta ABD $ is a right angled triangle,

\[\begin{align}

& \tan \left( \angle BAD \right)=\dfrac{BD}{AD} \\

& =\dfrac{10}{10} \\

& =1 \\

& \Rightarrow \angle BAD={{\tan }^{-1}}1 \\

& \Rightarrow \angle BAD=\dfrac{\pi }{4} \\

& \angle A=\angle CAD+\angle BAD \\

& =\dfrac{\pi }{3}+\dfrac{\pi }{4} \\

& =\dfrac{7\pi }{12} \\

& =105{}^\circ \\

\end{align}\]

Hence $\angle A$, as asked to us in the question = $\dfrac{7\pi }{12}\ or\ 105{}^\circ $.

Note: The question can also be solved using the cosine and sine formula after finding the length of hypotenuse of each triangle using Pythagoras Theorem.

Complete step by step answer:

In $\Delta ACD$, since $\Delta ACD$ is a right angled triangle,

$\begin{align}

& \tan \left( \angle CAD \right)=\dfrac{CD}{AD} \\

& =\dfrac{10\sqrt{3}}{10} \\

& =\sqrt{3} \\

& \Rightarrow \angle CAD={{\tan }^{-1}}\sqrt{3} \\

& \Rightarrow \angle CAD=\dfrac{\pi }{3} \\

\end{align}$

We will now find $\angle BAD$, since $\Delta ABD $ is a right angled triangle,

\[\begin{align}

& \tan \left( \angle BAD \right)=\dfrac{BD}{AD} \\

& =\dfrac{10}{10} \\

& =1 \\

& \Rightarrow \angle BAD={{\tan }^{-1}}1 \\

& \Rightarrow \angle BAD=\dfrac{\pi }{4} \\

& \angle A=\angle CAD+\angle BAD \\

& =\dfrac{\pi }{3}+\dfrac{\pi }{4} \\

& =\dfrac{7\pi }{12} \\

& =105{}^\circ \\

\end{align}\]

Hence $\angle A$, as asked to us in the question = $\dfrac{7\pi }{12}\ or\ 105{}^\circ $.

Note: The question can also be solved using the cosine and sine formula after finding the length of hypotenuse of each triangle using Pythagoras Theorem.

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