Help with two column proof of a trapezoid

May 2010
2
0
I have been asked to do a two column proof for a trapezoid. I am stumped on what to do. I am homeschooled and I am using a tough curriculum so the book I have for geometry assumes you know how to do everything or figure it out on your own. I would really appriciate some help please!!

figure,


question: Given: ROSE is a trapezoid with bases OS and RE.
Prove, OI • RI = EI • SI

and this is supposed to be a two column proof. :)

Thank you very very much!!
 
Dec 2009
3,120
1,342
I have been asked to do a two column proof for a trapezoid. I am stumped on what to do. I am homeschooled and I am using a tough curriculum so the book I have for geometry assumes you know how to do everything or figure it out on your own. I would really appriciate some help please!!

figure,


question: Given: ROSE is a trapezoid with bases OS and RE.
Prove, OI • RI = EI • SI

and this is supposed to be a two column proof. :)

Thank you very very much!!
You could begin by saying that the areas of triangles OSR and SOE are equal
as they have the same base and perpendicular heights.

This means that triangles OIR and SIE also have equal areas as they differ from the above-mentioned triangles
by the area of triangle OIS.

Now write the area of the triangle OIR using the angle at I

\(\displaystyle Area\ OIR=\frac{1}{2}|OI||IR|Sin(OIR)\)

Do the same for triangle SIE

\(\displaystyle Area\ SIE=\frac{1}{2}|SI||IE|Sin(SIE)\)

However, those angles are identical.

Hence the result follows
 
Nov 2007
985
175
Trumbull Ct
two column proof of a trapezoid

You could begin by saying that the areas of triangles OSR and SOE are equal
as they have the same base and perpendicular heights.

This means that triangles OIR and SIE also have equal areas as they differ from the above-mentioned triangles
by the area of triangle OIS.

Now write the area of the triangle OIR using the angle at I

\(\displaystyle Area\ OIR=\frac{1}{2}|OI||IR|Sin(OIR)\)

Do the same for triangle SIE

\(\displaystyle Area\ SIE=\frac{1}{2}|SI||IE|Sin(SIE)\)

However, those angles are identical.

Hence the result follows
Hello Archie and Minniemouse 2,

Archieyour proof is based on trigonometry and I believe would not be expected of a geometry student. It is aproblem involving parallel lines, transversals, equal angles and similar triangles.
Minnie if you do not understand these clues ask for more help.

bjh
 
Dec 2009
3,120
1,342
I have been asked to do a two column proof for a trapezoid. I am stumped on what to do. I am homeschooled and I am using a tough curriculum so the book I have for geometry assumes you know how to do everything or figure it out on your own. I would really appriciate some help please!!

figure,


question: Given: ROSE is a trapezoid with bases OS and RE.
Prove, OI • RI = EI • SI

and this is supposed to be a two column proof. :)

Thank you very very much!!
If a purely geometric solution is required,
you can use the following method....

Side OS is parallel to side RE.

\(\displaystyle |\angle{RIE}|=|\angle{OIS}|\)

\(\displaystyle |\angle{OSR|}=|\angle{OSI}|=|\angle{SRE}|=|\angle{IRE}|\)

\(\displaystyle |\angle{SOE}|=|\angle{SOI}|=|\angle{REO}|=|\angle{REI}|\)

Hence, triangles OSI and REI are equiangular.

Therefore triangle REI is a magnified version of triangle OSI.

Hence if we turn triangle OSI upside-down, we can compare corresponding sides, if we also turn it back to front.

Hence side OI corresponds to IE, SI corresponds to IR and SO corresponds to RE.

Each side of OSI is magnified by the same scale factor to become the corresponding side of triangle REI.

\(\displaystyle m|OI|=|IE|\ \Rightarrow\ m=\frac{|IE|}{|OI|}\)

\(\displaystyle m|SI|=|IR|\ \Rightarrow\ m=\frac{|IR|}{|SI|}\)

\(\displaystyle m|OS|=|RE|\ \Rightarrow\ m=\frac{|RE|}{|OS|}\)

\(\displaystyle \frac{|IE|}{|OI|}=\frac{|IR|}{|SI|}\ \Rightarrow\ |IE||SI|=|OI||IR|\)
 
Last edited:
  • Like
Reactions: bjhopper