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From the triangle below, if AD = 2 and CD= 8. Length of side AB

From the triangle below, if AD = 2 and CD= 8. Length of side AB-example-1
User Stackedup
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1 Answer

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Given: A triangle ABC with a point D on AC such that


\begin{gathered} AD=\text{ 2 units} \\ CD=8\text{ units} \\ \angle BDC=90\degree \end{gathered}

Required: To determine the length of side AB.

Step-by-step explanation: In triangle ABC, applying the Pythagoras theorem as


AC^2=AB^2+BC^2

Now,


\begin{gathered} AC=AD+CD \\ =10\text{ units} \end{gathered}

Thus,


100=AB^2+BC^2\text{ ...}(1)

Now, triangle ABD is a right-angled triangle with AB as the hypotenuse. Applying the Pythagoras theorem as-


\begin{gathered} AB^2=AD^2+BD^2 \\ AB^2-BD^2=4\text{ }(\because AD=2)...(2) \end{gathered}

Similarly, triangle BDC is a right-angled triangle. Thus-


\begin{gathered} BC^2=BD^2+CD^2 \\ BC^2-BD^2=64\text{ }(\because CD=8)\text{ ...}(3) \end{gathered}

Subtracting equations (2) and (3) as follows-


\begin{gathered} (AB^2-BD^2)-(BC^2-BD^2)=4-64 \\ AB^2-BC^2=-60\text{ }...(4) \end{gathered}

Again adding equations (1) and (4) as follows-


\begin{gathered} (AB^2+BC^2)+(AB^2-BC^2)=100+(-60) \\ 2AB^2=40 \\ AB^2=20 \\ AB=√(20)\text{ units} \end{gathered}

Further solving-


\begin{gathered} AB=2√(5)\text{ units} \\ AB=4.4721\text{ units} \\ AB\approx4.5\text{ units} \end{gathered}

Final Answer: Options (a) and (c) are correct.

User Belegnar
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