Question

Construct a triangle $ABC$ with the following data:

$AB=5\mathrm{cm},\mathrm{BC}=6\mathrm{cm}\mathrm{and}\angle \mathrm{ABC}=90°$

(i) Find a point$P$ which is equidistant from $B\mathrm{and}C$ and which is$5\mathrm{cm}$ from$A.$ How many such points are there?

(ii) Construct an inscribed circle of $∆ABC$ drawn above.

Answer 1

Step 1:

Draw a line segment$BC=6\mathrm{cm}$

Step 2:

At$B$, draw a ray$BX$making an angle of$90°$and cut off $BA=5\mathrm{cm}$.

Step 3:

Join $AC$.

Step 4:

Draw the perpendicular bisector of $BC$with the help of a compass, and with $B\mathrm{and}C$ as the center and with more than half of the line segment $BC$ as width, draw arcs above and below the line segment. and name these points as $P\mathrm{and}P\text{'}.$

Step 5:

From$A$, with radius $5\mathrm{cm}$draw arcs that intersect the perpendicular bisector $PP\text{'}$and name these point as $K$ and $L$ .

These are the two points.

Step 6:

Draw the angle bisector of $\angle B$intersecting $PP\text{'}$ at $O$ and name the point $D$ on $BC\mathrm{and}OD\perp BC$

Step 7: Construct an inscribed circle of $∆ABC$ drawn above.

Draw the angle bisectors of $\angle A\mathrm{and}\angle C$and mark this point as $O\text{'}$.

Step 8:

Now taking the radius $O\text{'}K$draw a circle that will touch the sides $AB\mathrm{and}AC$.

This is the required inscribed circle of $∆ABC$.

(i). Thus there are two such points K and L.

(ii). The required inscribed circle of $∆ABC$ is given below.