Q58 / 160 · TS EAMCET 2023 (14 May Shift 1) · Mathematics

The number of diagonals of a polygon is 35 . If A, B are two distinct vertices of this polygon, then the number of all those triangles formed by joining three vertices of the polygon having $\mathrm{AB}$ as one of its sides is

  • A
    $1$
  • Correct answer
    B
    $8$
  • C
    $10$
  • D
    $12$

Solution

Number of diagonals $={ }^n \mathrm{C}_2-n$ $$ \begin{aligned} & =\frac{n(n-1)}{2}-n=35 \\ & \Rightarrow n^2-n-2 n=35 \times 2 \Rightarrow n^2-3 n-70=0 \\ & \Rightarrow(n-10)(n+7)=0 ; \therefore n=10 \end{aligned} $$ To form triangle we need 3 vertices out of which 2 are $A$ and B $\therefore 8$ triangles can be formed by selecting 3 rd vertex as remaining 8 vertices of polygon.

Hence, the correct answer is option: 'B'.

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