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a) i)
Use the distance formula: d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)d=√(x2−x1)2+(y2−y1)2 to find the length OP,PQ,QR,PR, and OROP,PQ,QR,PR,andOR,
OP=sqrt((-5-0)^2+(4-0)^2)=sqrt(25+16)=sqrt41OP=√(−5−0)2+(4−0)2=√25+16=√41 m
PQ=sqrt((8+5)^2+(14-4)^2)=sqrt(13^2+10^2)=sqrt269PQ=√(8+5)2+(14−4)2=√132+102=√269 m
similarly, QR=sqrt97QR=√97 m, PR=sqrt290PR=√290 m, and
OR=sqrt169=13OR=√169=13 m
Use the law of cosines to find anglePQR∠PQR
PR^2=PQ^2+QR^2-2*PQ*QR*cosanglePQRPR2=PQ2+QR2−2⋅PQ⋅QR⋅cos∠PQR,
=> cosanglePQR=(PQ^2+QR^2-PR^2)/(2*PQ*QR)⇒cos∠PQR=PQ2+QR2−PR22⋅PQ⋅QR,
=> anglePQR=cos^-1((269+97-290)/(2*sqrt269*sqrt97))=76.394^@⇒∠PQR=cos−1(269+97−2902⋅√269⋅√97)=76.394∘
similarly,
anglePOR=cos^-1((OP^2+OR^2-PR^2)/(2*OP*OR))∠POR=cos−1(OP2+OR2−PR22⋅OP⋅OR)
=cos^-1((41+169-290)/(2*sqrt41*13))=118.720^@=cos−1(41+169−2902⋅√41⋅13)=118.720∘
area of DeltaPQR=A_1=1/2*PQ*QR*sinanglePQR
=1/2*sqrt269*sqrt97*sin76.394=78.50 " m"^2
area of DeltaPOR=A_2=1/2*OP*OR*sinanglePOR
=1/2*sqrt41*13*sin118.720=36.5 " m"^2
=> area of OPQR=A_1+A_2=78.5+36.5=115.00 " m"^2
a) ii)
Section formula :
If a point B(x,y) divides a line segment joining A(x_1,y_1)and C(x_2,y_2) in the ratio of m:n, i.e., (AB:BC=m:n),
then B(x,y)= ((mx_2+nx_1)/(m+n), (my_2+ny_1)/(m+n))
Given that B=(1,5) and C=(7,9) and C is 2 times from B, and 3 times from A,
=> AB:BC=1:2, => m:n=1:2
let coordinates of A=(x_1,y_1),
=> B(1,5)=((1xx7+2x_1)/(1+2),(1xx9+2y_1)/(1+2))
=> (x_1,y_1)=(-2,3)
Hence A(x_1,y_1)=A(-2,3)
