# triangle length with cosines

• May 6th 2008, 06:24 AM
jaynlisa08
triangle length with cosines
Hi guys just need some help.

being given this and havent got a clue.

if you could work it out and then explain that would be great

Points P+Q are on opposite sides of a straight tunnel. A third point R forms a triangle PQR. S+T are established along the sides PR and QR repectively such that

PR= 312m
SR= 116m
ST= 275m
QR= 423m
RT= 287m

I need to calculate the length of the tunnel assuming that all points are in the same place.

Thanks PPL

James
• May 6th 2008, 08:33 AM
Soroban
Hello, jaynlisa08!

Quote:

Points P and Q are on opposite sides of a straight tunnel.
A third point R forms a triangle PQR.
S and T are on sides PR and QR repectively such that :

PR= 312m, SR= 116m, ST= 275m, QR= 423m, RT= 287m

Find the length of PQ.

Code:

R
o
*  *
*    *
*        *
S o * * * * * o T
*              *
*                *
*                    *
P o * * * * * * * * * * * o Q

In $\displaystyle \Delta PRT$ use the Law of Cosines:

. . $\displaystyle \cos R \:=\:\frac{RS^2 + RT^2 - ST^2}{2(RS)(RT)} \;=\;\frac{116^2 + 287^2 - 275^2}{2(116)(287)} \;=\;\frac{2525}{8323}$

In $\displaystyle \Delta PRQ$, use the Law of Cosines:

. . $\displaystyle PQ^2 \;=\;PR^2 + QR^2 - 2(PR)(QR)\cos R$

. . $\displaystyle PQ^2\;=\;312^2 + 423^2 - 2(312)(423)\,\frac{2525}{8323} \;= \;198,089.3162$

Therefore: .$\displaystyle PQ \;\approx\;442.94$ m.