Triangles

[rktpro]

This problem was in a book.

In the figure, angle RQS = angle RSQ and QS is the median on TR. Prove that: EP/TP = PQ/TS

Our teacher said that this problem is incomplete. Also, when you draw the figure, we need angle PTS for construction. But this angle is not given. Thus, as I find, the problem seems wrong.

Can you help me solve it if you know how to?

Thanks.

Edited September 7, 2011 by rktpro

[imatfaal]

Seems screwy to me. Is that the diagram provided? - cos it is deliberately misleading if it is. RSQ does not look at all equal to SQR - but does look equal to QRS.

 

I think the problem is incomplete - but I have a feeling (just a guess I havent tried) you could make a stab at it with the above correction and the assumption that EP is perpendicular to RT

[rktpro]

Seems screwy to me. Is that the diagram provided? - cos it is deliberately misleading if it is. RSQ does not look at all equal to SQR - but does look equal to QRS.

 

I think the problem is incomplete - but I have a feeling (just a guess I havent tried) you could make a stab at it with the above correction and the assumption that EP is perpendicular to RT

 

Go for the given stuff only. The diagram is by me on MS Paint and not too detailed.

Yes, EP must be prependicular to RT, but that is not given. This is where I am confused too.

[imatfaal]

Prob better if you put on forum everything that was given - and how it was given.

[rktpro]

Prob better if you put on forum everything that was given - and how it was given.

 

Do you make figures by scale when you are given a statement. Well, we just draw a rough one. Besides, even if I correct the figure, I don't know about any other angles and thus it would be impossible.

How to get result with just what is given?

[imatfaal]

I seem to be the only one replying - and before I spend anytime really thinking about it I would like to know exactly what was asked - the devil is sometimes in the details

[rktpro]

I seem to be the only one replying - and before I spend anytime really thinking about it I would like to know exactly what was asked - the devil is sometimes in the details

 

How many times would I have to say that this is the statement of the question. Nothing else is given!

In the figure, angle RQS = angle RSQ and QS is the median on TR. Prove that: EP/TP = PQ/TS

[imatfaal]

I cannot work this out - I am getting the ratios being 1 apart, not equal

 

assuming RT is perp to EP - and assuming angle RQS and angle RSQ lie between 45 and 90

 

1. angle RQS and RSQ both equal a

2. by opposite angles EST also equals a

3. call angle SET b - and bisect angle QRS and join to median point of QS - call new point M

4. triangles EST and RMQ and RMS are similar (for right-angled triangle only need one other angle to show similarity)

5. angle RQM and angle RSM must be same and equal to b as triangles similar.

6. angle QRS is sum of angle RQM and angle RSM ie 2b

7. call length TS and SR J - note this must also be RQ as triangle QRS must be isoceles

 

PT/TE = PQ/TS

8. PT. tan(2b) = o/a = PT/2J. PT = 2J Tan(2b)

9. TE. tan(a) = o/a = TE/2J. TE=J Tan(a) . But as a+b+90=180 (see triangle EST) then a= 90-b. As tan(90-x) = 1/tan(x) then TE = J/Tan(b)

10.PQ. cos (2b)=a/h = 2J/PR. PR= 2J/Cos(2b). PQ = PR - J = 2J/Cos(2b) - J

11. TS defined as J

 

[math] \frac{2J.tan(2b)}{\frac{J}{tan(b)}} = \frac{\frac{2J}{cos(2b)}-J}{J} [/math]

 

but everytime I simplify I end up with the RHS being 1 bigger than than the LHS

[TonyMcC]

I think there may be a misprint in the original question.

First let me say I have not worked it through completely. However it looks to me as if there would be more chance of proving EP/TP= PR/TS.

I say this because triangle QRS is isosceles making QR=SR.

Since QS is the median 0n TR, TS=SR=QR.

Thus, it seems to me, that the objective might be prove EP/TP= PR/TS (TS being equal to QR).

I don't think it is necessary for EP to be perpendicular to RT.

Edited September 13, 2011 by TonyMcC

[imatfaal]

Tony

Just thinking quickly about your variant (in a numerical fashion) to see if it might be generalisable - I subbed in angle QSR as 60 degrees (makes everything simple and triangles all similar). EP/TP would equal 3/2 but PR/TS would equal 4/1.

 

 

Just thought to run through the original question with 60 degrees and I still get a variation of the ratios by 1 - I reckon you should be trying to prove that PT/TE = PQ/TS -1

Edited September 13, 2011 by imatfaal

[rktpro]

I cannot work this out - I am getting the ratios being 1 apart, not equal

 

assuming RT is perp to EP - and assuming angle RQS and angle RSQ lie between 45 and 90

 

1. angle RQS and RSQ both equal a

2. by opposite angles EST also equals a

3. call angle SET b - and bisect angle QRS and join to median point of QS - call new point M

4. triangles EST and RMQ and RMS are similar (for right-angled triangle only need one other angle to show similarity)

5. angle RQM and angle RSM must be same and equal to b as triangles similar.

6. angle QRS is sum of angle RQM and angle RSM ie 2b

7. call length TS and SR J - note this must also be RQ as triangle QRS must be isoceles

 

PT/TE = PQ/TS

8. PT. tan(2b) = o/a = PT/2J. PT = 2J Tan(2b)

9. TE. tan(a) = o/a = TE/2J. TE=J Tan(a) . But as a+b+90=180 (see triangle EST) then a= 90-b. As tan(90-x) = 1/tan(x) then TE = J/Tan(b)

10.PQ. cos (2b)=a/h = 2J/PR. PR= 2J/Cos(2b). PQ = PR - J = 2J/Cos(2b) - J

11. TS defined as J

 

[math] \frac{2J.tan(2b)}{\frac{J}{tan(b)}} = \frac{\frac{2J}{cos(2b)}-J}{J} [/math]

 

but everytime I simplify I end up with the RHS being 1 bigger than than the LHS

 

That went above my head. The theorem that has to be used is Basic Proportionality Theorem bacause it is in the syllabus. But, I assume you to be correct here.

 

I think there may be a misprint in the original question.

First let me say I have not worked it through completely. However it looks to me as if there would be more chance of proving EP/TP= PR/TS.

I say this because triangle QRS is isosceles making QR=SR.

Since QS is the median 0n TR, TS=SR=QR.

Thus, it seems to me, that the objective might be prove EP/TP= PR/TS (TS being equal to QR).

I don't think it is necessary for EP to be perpendicular to RT.

 

Yes, I had thought of it too. It could be corrected as EP/TP=PR/QR too. The point here is, if I join TQ and ER, I still wouldn't get TQ parallel to ER. Try by Basic Proportionality theorem.