If #hatQ# is a Right-angle in #DeltaPQR and PR-PQ=9 , PR-QR=18# then what is the perimeter of the triangle?

2 Answers
Mar 18, 2018

Perimeter is #108#

Explanation:

As in right angled triangle #PQR#, #m/_Q=90^@#, #bar(PR)# is hyptenuse and te largest side. Further, according to Pythagoras theorem

#PR^2=QR^2+PQ^2# .......................(A)

Now let #PR=x#, then as #PR-PQ=9#, #PQ=x-9#

and as #PR-QR=18# and therefore #QR=x-18#

Putting these values in (A), we have

#x^2=(x-9)^2+(x-18)^2#

or #x^2=x^2-18x+81+x^2-36x+324#

or #x^2-54x+405=0#

or #x^2-45x-9x+405=0#

or #x(x-45)-9(x-45)=0#

or #(x-9)(x-45)=0#

Observe that as all sides are positive, we must have #x>18#

and hence only solution is #x=45#

and sides are #45#, #36# and #27#

and perimeter is #45+36+27=108#

Mar 18, 2018

Perimeter = #27+36+45 = 108#

Explanation:

If #hatQ# is #90°#, then #PR# is the hypotenuse.

#PQ and RQ# can be written in terms of the hypotenuse.

Let #PR = x#

Then #PQ = x-9 and RQ = x-18#

Write an equation using Pythagoras' Theorem:

#(x-9)^2 +(x-18)^2 = x^2#

#x^2 -18x+81 + x^2 -36x+324= x^2#

#x^2 -54x + 405 =0#

#(x-45)(x-9)=0#

#:. x=45 or x =9 rarr# reject #9# as being too short

If the hypotenuse is #45#, then the sides are

#45-9 = 36 and 45-18 = 27#

Perimeter = #27+36+45 = 108#