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Question Number 24179 by gopikrishnan005@gmail.com last updated on 14/Nov/17

the line segment joining the points (3,−4) and (1,2) is trisected at the points P and Q if the coordinates of P and Q are (p,−2) and (5/3,q) respectively find the values of p and q.

$${the}\:{line}\:{segment}\:{joining}\:{the}\:{points}\:\left(\mathrm{3},−\mathrm{4}\right)\:{and}\:\left(\mathrm{1},\mathrm{2}\right)\:{is}\:{trisected}\:{at}\:{the}\:{points}\:{P}\:{and}\:{Q}\:{if}\:{the}\:{coordinates}\:{of}\:{P}\:{and}\:{Q}\:{are}\:\left({p},−\mathrm{2}\right)\:{and}\:\left(\mathrm{5}/\mathrm{3},{q}\right)\:{respectively}\:{find}\:{the}\:{values}\:{of}\:{p}\:{and}\:{q}. \\ $$

Answered by mrW1 last updated on 14/Nov/17

((x_1 −3)/(1−3))=(1/3) ⇒x_1 =3−(2/3)=(7/3) ((y_1 −(−4))/(2−(−4)))=(1/3) ⇒y_1 =2−4=−2 ((x_2 −1)/(3−1))=(1/3) ⇒x_2 =1+(2/3)=(5/3) ((y_2 −2)/(−4−2))=(1/3) ⇒y_2 =2−2=0 ⇒p=(7/3) ⇒q=0

$$\frac{{x}_{\mathrm{1}} −\mathrm{3}}{\mathrm{1}−\mathrm{3}}=\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\Rightarrow{x}_{\mathrm{1}} =\mathrm{3}−\frac{\mathrm{2}}{\mathrm{3}}=\frac{\mathrm{7}}{\mathrm{3}} \\ $$$$\frac{{y}_{\mathrm{1}} −\left(−\mathrm{4}\right)}{\mathrm{2}−\left(−\mathrm{4}\right)}=\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\Rightarrow{y}_{\mathrm{1}} =\mathrm{2}−\mathrm{4}=−\mathrm{2} \\ $$$$ \\ $$$$\frac{{x}_{\mathrm{2}} −\mathrm{1}}{\mathrm{3}−\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\Rightarrow{x}_{\mathrm{2}} =\mathrm{1}+\frac{\mathrm{2}}{\mathrm{3}}=\frac{\mathrm{5}}{\mathrm{3}} \\ $$$$\frac{{y}_{\mathrm{2}} −\mathrm{2}}{−\mathrm{4}−\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\Rightarrow{y}_{\mathrm{2}} =\mathrm{2}−\mathrm{2}=\mathrm{0} \\ $$$$ \\ $$$$\Rightarrow{p}=\frac{\mathrm{7}}{\mathrm{3}} \\ $$$$\Rightarrow{q}=\mathrm{0} \\ $$

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