Question

Construct each of the following and give justification: A triangle $PQR$ given that $QR=3\mathrm{cm},\angle PQR={45}^{°},\mathrm{and}QP-PR=2\mathrm{cm}$.

Answer 1

Step 1. Draw a ray $OX$ and a line segment $QR=3\mathrm{cm}$ from it.

Step 2. Construct an angle $\angle YQR={45}^{°}$ at point $Q$. Cut off $QS=2\mathrm{cm}$ from the line segment $QY$ and join the point $R$ and $S$.

Step 3. To meet $QY$ at $P$, draw a perpendicular bisector of $RS$.

Hence, the required graph is shown below:

Make a justification.

$\angle PQR$ and base $QR$ are drawn.

$P$ lies on the perpendicular bisector of $SR$ as shown in the construction above.

$PS=PR\cdots \left(1\right)$ [From Construction]

From the construction,

$\begin{array}{rcl}QP& =& QS+PS\\ & \Rightarrow & QS=QP-PS\\ & \Rightarrow & QS=QP-PR\left[\mathrm{from}\left(1\right)\right]\end{array}$

Hence, the construction is justified.