pyth. theorm 的問題

a pinting in an gallery is hung by a wire of lenght 42cm at points P and Q on a nail.



the picture is hung in such a way that PR and QR are perpendicular to each other, wher PR is shorter than RQ.
a)Given than PR= x cm, express QR in tern of x
b)Given that the lenght of the painting is 30cm, user Pythagoras' Theorem to form an equation in x and show that it reduce to to x*x-42x+432=0
c)slove the equation for x and find the area of triangle PQR.
Since you have a picture, I do not draw it again
(a)
PR+RQ=42 ( lenght of a wire 42cm)
RQ=42-PR=(42-x) cm
(b)
By pyth. theorm
PQ^2=PR^2+RQ^2
30^2=x^2+(42-x)^2 (Because the lenght of the painting is 30cm)
900=x^2+1764-84x+x^2
2x^2-84x+864=0
x^2-42x+432=0
(c)
x^2-42x+432=0
(x-24)(x-18)=0
x=24 or 18
Since PR is shorter than RQ
x<42-x
2x<42
x<21
So x=18
PR=18 cm
QR=42-18=24 cm
The area of triangle PQR
=PR*QR/2
=18*24/2
=216 cm^2