Question

Construct $△$ PYQ such that, PY $=6.3cm,YQ=7.2cm,PQ=$ $5.8cm.$ If $=YZYQ=65$ then construct $△XYZ$ similar to $△PYQ$

Answer 1

Analysis:
As shown in the figure,
Let $Y-Q-Z$ and $Y-P-X$.
$△XYZ\sim △PY{Q}_{\dots }$ [Given]
$\therefore \angle XYZ\cong \angle PYQ\dots$ [ [Corresponding angles of similar triangles]
$\frac{XY}{PY}=\frac{YZ}{YQ}=\frac{XZ}{PQ}\dots$ (i) [Corresponding sides of similar triangles]
But, $\frac{YZ}{YQ}=\frac{6}{5}$, ,.. (ii) [Given $]$
$\therefore \frac{XY}{PY}=\frac{YZ}{YQ}=\frac{XZ}{PQ}=\frac{6}{5}\dots [$ From (i) and (ii)]
$\therefore$ sides of $△XYZ$ are longer than corresponding sides of $△$ $PYQ$.
$\therefore$ If $\mathrm{seg}YQ$ is divided into 5 equal parts, then seg $YZ$ will be 6 times each part of ${\mathrm{seg}}_{.}YQ$.

So, if we construct $△PYQ$, point $Z$ will be on side $YQ$, at a distance equal to 6 parts from $Y$.

Now, point $X$ is the point of intersection of ray $YP$ and a line through $Z$, parallel to $PQ$.
$△XYZ$ is the required triangle similar to $△PYQ$.
Steps of construction:
i. Draw $△$ PYQ of given measure. Draw ray YT making an acute angle with side YQ.
ii. Taking convenient distance on compass, mark 6 points $Y_1$, $Y_2,Y_3,Y_4,Y_5$ and $Y_6$ such that
$Y_1=Y_1{Y}_2=Y_2{Y}_3=Y_3{Y}_4=Y_4{Y}_5=Y_5{Y}_6$.
iii. Join $Y_{5}Q$. Draw line parallel to $Y_{5}Q$ through $Y_6$ to
intersects ray $YQ$ at $Z$.
iv. Draw a line parallel to side $PQ$ through $Z$. Name the point of intersection of this line and ray YP as $X_s$
$△XYZ$ is the required triangle similar to $△PYQ$.