Question

Construct a triangle $ABC$ in which $BC=7\text{ cm}$ , $\angle B={{75}^{0}}$ and $AB+AC=13\text{ cm}$ .

Answer 1

Hint: For solving this question first we will see how to draw an angle of ${{75}^{0}}$ , and how to draw perpendicular bisector of any line segment. After that, we will proceed stepwise as per the data given in the problem to draw the triangle ABC as per the given configurations and know how to draw angles and lines using ruler and compass.

Complete step-by-step solution -
Given:
We have to construct a triangle $ABC$ in which $BC=7\text{ cm}$ , $\angle B={{75}^{0}}$ and $AB+AC=13\text{ cm}$ .
Now, before we proceed we should know how to draw an angle of ${{75}^{0}}$ , and how to draw perpendicular bisector of any line segment.
Follow following steps to draw $\angle BAC={{75}^{0}}$ :
1. Draw a segment of length 7 cm and name it AB. The figure is shown below:

2. Now, draw an arc of a certain radius less than 7 cm with centre at point A as shown in the figure below:

3. Now, draw an arc of the same radius with centre at point C and make it intersect the previous one at point G. The figure is shown below:

4. Now, draw an arc of the same radius with centre at point G and make it intersect the previous one at point J. The figure is shown below:

5. Now, draw an arc of the same radius with centre at point J and make it intersect the previous one at point L and join AL and segment AL will intersect the arc CD at point M. The figure is shown below:

6. Now, draw an arc of the same radius with centre at point M and make it intersect the arc CH at point O and join AO and make it intersect the arc CD at point P. The figure is shown below:

7. Now, $\angle OAB$ is the required angle of ${{75}^{0}}$ .
Follow the following steps to draw perpendicular bisector of any line segment:
1. Let there be a line AB of a certain length as shown in the figure below:

2. Now, draw an arc of radius more than half length of the segment AB with centre A on the above and below of the segment as shown in the figure below:

3. Now, draw an arc of the same radius with centre B on the above and below of the segment and make it intersect the previous two arcs as shown in the figure below:

4. Now, join the point K and point L as shown in the figure below:

5. Now, KL is the required perpendicular bisector of the line segment AB.
Now, we will use these methods to draw the diagram as per the given data. Follow the following steps to draw the required $\Delta ABC$ :
1. Draw a segment of length 7 cm and name it BC. The figure is shown below:

2. Now, with the help of compass draw an angle of 75 degrees as shown in the figure below:

3. Now, draw an arc of radius 13 cm with centre at point B and make it intersect BX at point F and join CF as shown in the figure below:

4. Now, draw the perpendicular bisector of the FC and make it intersect segment FB to get point A and join AC as shown in the figure is shown below:

5. Now, ABC is the required triangle and we can verify the given condition $AB+AC=13\text{ cm}$ by measuring AB and AC with the ruler as shown in the figure below:

6. Now, from the above figure, we conclude that the length of side AB will be 5.4 cm and the length of the side AC will be 7.6 cm. Then,
$\begin{align}
  & AB+AC=5.4+7.6 \\
 & \Rightarrow AB+AC=13 \\
\end{align}$
Hence, verified.

Note: Here, the student must draw the diagram neatly and correctly. Moreover, while drawing the diagram student should be aware of what is asked in the question and proceed as per the instructions given to get the correct figure of the triangle ABC as per the given configurations in the end.