Question

Two circles of radii 3cm and 2cm have their centres 9cm apart. Draw a transverse common tangent and measure the length of the tangent and write the measurement.

Answer 1

Hint:Draw a line segment AB of length 9cm. At the endpoints of the line segment, draw two circles of radii 3cm and 2cm with centres at A and B respectively. To draw the transverse common tangent of the two circles, draw a perpendicular bisector of line segment and another circle of radius equal to the sum of radii of two smaller circles with centre as one of the endpoints of the line segment. Draw tangent from the other endpoint of the line segment to the bigger circle and then draw the transverse common tangent. Measure its length using a scale and pencil.

Complete step-by-step answer:
We have two circles of radii 3cm and 2cm whose centres are 9cm apart. We have to draw a transverse common tangent to the circles and measure its length.
To do so, we will first draw a line segment AB of length 9cm using pencil and scale.

We will now draw two circles of radii 3cm and 2cm with centres at A and B respectively. To do so, we will measure 3cm and 2cm on the compass using a scale and taking A and B as centre, drawing the circles.

We will now draw a perpendicular bisector of line AB. To do so, open compass more than half of the distance between A and B and scribe arcs of the same radius above and below the line segment AB with centres at A and B.

We will label the point of intersection of arcs as M and N. We will join the line segment MN using scale and pencil. We will label the point of intersection of AB and MN as C.

We will draw a circle with the centre at A and of radius equal to the sum of radii of two small circles, i.e., the radius of this circle is $=3+2=5cm$.

Taking C as centre and AC as radius, we will scribe an arc on the circle of radius 5cm. We will label the point of intersection of arc and circle as P. We will join the line segment BP.

We will now join line segment AP and label the intersection of this line segment with the circle of radius 3cm as Q.

We will draw a line segment BR, with R at the circle of radius 2cm such that BR is parallel to PQ.

We will join the line segment RQ. This is the common transverse tangent of both the circles.

We will now measure the length of the line segment QR using scale and pencil.
Thus, the measure of QR is $7.48cm$.
Hence, the length of the transverse common tangent of circles with radii 3cm and 2cm is $7.48cm$.

Note: We can verify the length of transverse tangent of both the circles by using the formula $\sqrt{{{d}^{2}}-{{\left( {{r}_{1}}+{{r}_{2}} \right)}^{2}}}$, ‘d’ is the distance between centres of two circles and ${{r}_{1}},{{r}_{2}}$ are radii of the circles. One must keep in mind that all the lines should be straight and must be drawn with a pencil and scale. It’s necessary to label all the vertices and measurements of line segments during each step of construction.