Construction of a triangle

Triangle construction refers to the process of drawing a triangle using specific tools and steps in geometry. By following a set of procedures, we can create a triangle with accurate measurements and geometric properties. We use tools like a compass, ruler, and protractor to ensure precision.

To construct a triangle, we start with given specifications such as side lengths or angles. We then use the tools to draw lines, angles, and arcs based on these specifications. Making measurements and connecting specific points or intersections allows us to form the sides and angles of the triangle.

Triangles can be constructed based on various postulates. Here are some common postulates for constructing triangles:

1. Side-Side-Side (SSS): Given the lengths of all three sides of a triangle, the triangle can be constructed using these specified side lengths.

2. Side-Angle-Side (SAS): Given the length of two sides and the measure of the included angle, the triangle can be constructed by drawing the specified sides and connecting them with the given angle.

3. Angle-Side-Angle (ASA): Given the measure of two angles and the length of the side included between them, the triangle can be constructed by drawing the specified angles and connecting them with the given side.

4. Angle-Angle-Side (AAS): Given the measure of two angles and the length of a side not included between them, the triangle can be constructed by drawing the specified angles, drawing the given side, and connecting them.

5. Side-Angle-Angle (SAA): If the length of one side and the measures of two angles not included between the sides are given, it is not possible to construct a unique triangle. Multiple triangles with different shapes can be formed in this case.

Let’s learn how to construct triangles based on above postulates

Two sides and included angle are given

Construct a triangle ABC in which $\overline{AB} = 4.8 cm,$ $\overline{AC} = 3.2 cm,$ and $m\angle B = 75\degree$

Steps of Construction:

1. Draw a line segment $\overline{AB}$ of $4.8 cm$ in length.

   

2. Using a protractor, mark a $75\degree$ angle from point $A$, labeling it as point $C$.

   

3. With a compass, set the width to $3.2 cm$ and draw an arc from point $A$ intersecting line segment $\overline{AB}$ at point $C$.

   

4. Connect points $B$ and $C$ with a straight line segment.

$\triangle ABC$ is the required triangle.

One side and two angles are given

Construct a triangle ABC in which $\overline{AB} = 5.5 cm,$ $m\angle A = 75\degree$ and $m\angle B = 40\degree$

Steps of Construction:

1. Draw a line segment $\overline{AB}$ of $5.5 cm$.

   

2. Place the protractor's origin at point $A$ and ensure that its base line coincides with line $\overline{AB}$.

   

3. Locate the $75\degree$ mark on the protractor and mark a point, then extend ray from point $A$ to the mark.

   

4. Move the protractor to point $B$ and align its base line again with line $\overline{AB}$.

   

5. Locate the $40\degree$ mark on the clockwise scale of the protractor and mark a point, extend the ray from point $B$ to marked point.

   

6. Connect the intersection point of the two rays with point $B$, labeling it as point $C$.

$\triangle ABC$ is the required triangle.

Two sides and opposite angle are given

Construct a triangle ABC in which $\overline{PR} = \overline{QR} = 4.7 cm$ and $m\angle P = 55\degree$

Steps of Construction:

1. Draw a line segment $\overline{PR}$ of $4.7 cm$.

   

2. Place the protractor's origin at point $P$ and ensure that its base line coincides with line $\overline{PR}$.

   

3. Locate the $55\degree$ mark on the protractor and mark a point on the ray extending from point $P$.

   

4. With a compass, set the width to $4.7 cm$ and draw an arc from point $R$ intersecting line segment $\overline{PQ}$ at point $Q$

   

5. Join points $Q$ and $R$ $\triangle PQR$ is the required triangle.