Question

How to show?!

PQR is an equilateral triangle of side length `x`, and PS is the perpendicular from P to QR. PS is produced to T so that PT = `x`.

A. Show that angle PQT = 75◦ and hence that angle SQT = 15◦.

B. Show that QS = `1/2 x` and PS = `1/2 x sqrt(3)`

C. Show that ST = `1/2 x (2 - sqrt(3))`

D. Hence show that that tan15◦ = 2 − `sqrt(3)`

Answer 1

PQR is an equilateral triangle of side length `x`, and PS is the perpendicular from P to QR. PS is produced to T so that PT = `x`.

A. Show that angle PQT = 75◦ and hence that angle SQT = 15◦.

B. Show that QS = `1/2 x` and PS = `1/2 x sqrt(3)`

C. Show that ST = `1/2 x (2 - sqrt(3))`

D. Hence show that that tan15◦ = 2 − `sqrt(3)`

Given `PQR` is equilateral and `PQ=QR=RP=x`

`PS_|_RQ`

So `DeltaPQS~=DeltaPRS`

Hence `QS=SR=1/2QR=1/2x`
And
`angleQPS=angle RPS=1/2angleQPR=1/2*60^@=30^@`

In `DeltaQPT,QP=PT`

So `anglePQT=angle PTQ=1/2(180^@-angleQPT)`

`=1/2(180^@-30^@)=75^@`

Now `angleSQT=angle PQT-anglePQS=75^@-60^@=15^@`

Now `QS=1/2x`
And
`PS=sqrt(PQ^2-QS^2)=sqrt(x^2-x^2/4)=sqrt3/2x`

So `ST=PT-PS=x-sqrt3/2x=x/2(2-sqrt3)`

Now

`tan15^@=tanangleSQT=(ST)/(QS)=2-sqrt3`