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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The value of the length DF is 1/4 of that of CD because of the Angle Bisector Theorem for triangles.
<h3>What is the Angle Bisector Theorem?</h3>
The Angel Bisector Theorem states that if a ray bisects an angle of a triangle, it dissects the side of the triangle that is opposite to the dissected angle into parts that are proportionate to the adjacent sides.
Note that the angle that has been dissected here is ∠CBD. Thus,
=
See the link below for more about Angle Bisector Theorem of the Triangle:
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