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In a polygon, no three diagonals areconcurrent. If the total number ofpoints of intersection of diagonals interior to the polygon is 70, then thenumber of diagonals of the polygon isa. `20`b. `28`c. `8`d. none of theseA. 8B. 20C. 28D. none of these |
Answer» Let there be n-sides of the polygon. Then, number of vertivces is also n. We know that in a convex polygon every group of four vertices determines two diagonals and hence one point of intersection So. Total numbers of points of interesction of diagonals is `""^(n)C_(4)`. `:.""^(n)C_(4)=70` `impliesn(n-1)(n-2)(n-3)=70xx24` `impliesn(n-1)(n-2)(n-3)=5xx6xx7xx8` `impliesn=8` Hence, number of diaonals `=""^(n)C_(2)-n=""^(8)C_(2)-8=20`. |
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