202207071711 Solution to 1976-CE-AMATH-I-XX

Find the ratio in which the line segment joining the points $P(-2,3)$ and $Q(3,5)$ is divided by the $y$ -axis. Find also the $y$ -coordinate of the point of division.

Preparation.

i. Set up a Cartesian coordinate system on a grid-lined graph paper:

ii. Plot the origin $O(0,0)$ , point $P(-2,3)$ , and point $Q(3,5)$ :

iii. Join all three points with segments $OP$ , $OQ$ , and $PQ$ :

iv. Target at the $y$ -intercept of line segment $PQ$ :

v. By the built-in function Properties... of The Geometer’s Sketchpad,

we are half done with $y\textrm{-intercept}=3.8$ .

vi. Next, what is the ratio $PE:EQ$ ?

$1^\circ$ :

$2^\circ$ :

Numerically,

$\begin{aligned} PE:EQ & = \frac{PE}{EQ} \\ & = \frac{2.15}{3.23} \\ & = 0.666\qquad \textrm{(3 s.f.)} \\ \end{aligned}$

for

Analytically,

for points $P(-2,3)$ and $Q(3,5)$ on the Cartesian plane with origin $O(0,0)$ , write vectors $\mathbf{OP}=-2\,\hat{\mathbf{i}}+3\,\hat{\mathbf{j}}$ and $\mathbf{OQ}=3\,\hat{\mathbf{i}}+5\,\hat{\mathbf{j}}$ , so that

$\begin{aligned} \mathbf{PQ} & =\mathbf{PO}+\mathbf{OQ} \\ & = -\mathbf{OP}+\mathbf{OQ} \\ & = -\Big( -2\,\hat{\mathbf{i}}+3\,\hat{\mathbf{j}}\Big) + \Big( 3\,\hat{\mathbf{i}}+5\,\hat{\mathbf{j}} \Big) \\ & =5\,\hat{\mathbf{i}}+2\,\hat{\mathbf{j}} \\ \end{aligned}$