First slide

Properties of triangles

Question

$In a triangle the length of the two$ $larger sides are 10 and 9,$ $r e s p e c t i v e l y . I f t h e a n g l e s a r e i n A . P ., t h e n t h e l e n g t h o f t h e t h i r d s i d e c a n b e :$

Moderate

A

$\sqrt{91}$
B

$3 \sqrt{3}$
C

5
D

None of these

Solution

$\begin{array}{l} Since the angles of a triange are in AP \\ let the third side be x \\ \Rightarrow (α - δ) + α + (α + δ) = 180^{°} \Rightarrow α = 60^{°} \\ Use cosine law in △ A B C \\ \cos 60^{°} = \frac{{(10)}^{2} + {(x)}^{2} - 9^{2}}{2 \cdot (10) \cdot (x)} \Rightarrow \frac{1}{2} = \frac{19 + x^{2}}{20 x} \\ \Rightarrow x^{2} - 10 x + 19 = 0 \\ x = \frac{10 \pm \sqrt{100 - 76}}{2} = 5 \pm \sqrt{6} \end{array}$

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