Question

State whether the following statement is true or false.

Enter $1$ for true and $0$ for false

Point P (5, 3) is one of the two points of trisection of the line segment joining the points A (7, 2) and B (1, 5).

Answer 1

$AssumingthatPliesonAB,$

$WearetoinvestigateifthepointP(x_1,y_1)=(5,3)trisectsthe$

$linesegmentjoiningthepointsA(x_2,y_2)=(7,2)\&B(x_3,y_3)=(1,5).$

$\therefore AP:BPshouldbe=1:2.$

$WeobtainthelengthsAP\&BPbyapplyingthedistance$

$formulad=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}.$

$SoAP=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}=\sqrt{{(7-5)}^2+{(2-3)}^2}=\sqrt{5}and$

$BP=\sqrt{{(x_3-x_1)}^2+{(y_3-y_1)}^2}=\sqrt{{(1-5)}^2+{(5-3)}^2}=\sqrt{20}.$

$\therefore AP:BP=\sqrt{5}:\sqrt{20}=1:2$

$SoPdividesABintheratio1:2.$

$i.ethestatementistrue.$

$Ans-TRUE.$