Show that it is not possible for the lengths of the
1 answer:
1. Check the picture below.
2. By the well known intersecting secants property: AO*OC=BO*OD
3. Since the segments are assumed to be consecutive integers, AO, OC cannot be the smaller, and neither can AO<BO and OC<OD.
The only possibility is that the multiplication of the smallest to the largest is equal to the multiplication of the middle 2.
4. For example : 4*5=2*10, we must have 4>2 but 5<10
Here the ordering is 2<4<5<10, similarly assume BO<AO<CO<DO
5. since the segments are of consecutive integers length, let them be
BO=n, AO=n+1, CO=n+2, DO=n+3
6. AO*CO=BO*DO
so n(n+3)=(n+1)(n+2)
so 0=3 which makes no sense, so no solution with the assumed conditions.
2. By the well known intersecting secants property: AO*OC=BO*OD
3. Since the segments are assumed to be consecutive integers, AO, OC cannot be the smaller, and neither can AO<BO and OC<OD.
The only possibility is that the multiplication of the smallest to the largest is equal to the multiplication of the middle 2.
4. For example : 4*5=2*10, we must have 4>2 but 5<10
Here the ordering is 2<4<5<10, similarly assume BO<AO<CO<DO
5. since the segments are of consecutive integers length, let them be
BO=n, AO=n+1, CO=n+2, DO=n+3
6. AO*CO=BO*DO
so n(n+3)=(n+1)(n+2)
so 0=3 which makes no sense, so no solution with the assumed conditions.
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