Construction of triangle given SSS, SAS, ASA — step by step with compass

Question

Construct a triangle with the following measurements using a ruler and compass:

SSS: Sides 5 cm, 4 cm, and 3 cm
SAS: Two sides 6 cm and 4 cm with an included angle of 60°
ASA: Two angles 45° and 75° with the included side 5 cm

(NCERT Class 7 — Practical Geometry)

Solution — Step by Step

Construction 1: SSS (Sides 5, 4, 3 cm)

Draw a line segment $BC = 5$ cm using a ruler. This is our base — always start with the longest side for a neat diagram.

Open the compass to 4 cm. Place the pointed end at $B$ and draw an arc above $BC$ . Now open the compass to 3 cm, place it at $C$ , and draw another arc. The point where these two arcs intersect is $A$ .

Join $A$ to $B$ and $A$ to $C$ . Triangle $ABC$ with sides 5, 4, and 3 cm is ready. Notice this is actually a right triangle ( $3^2 + 4^2 = 5^2$ ).

Construction 2: SAS (6 cm, 60°, 4 cm)

Draw $BC = 6$ cm. At point $B$ , construct an angle of 60° using a compass (standard 60° construction: arc from $B$ , then same radius arc from the intersection point).

Along the 60° ray from $B$ , mark point $A$ at 4 cm from $B$ .

Join $A$ to $C$ . Triangle $ABC$ with $BC = 6$ cm, $BA = 4$ cm, and $\angle B = 60°$ is done.

Construction 3: ASA (45°, 5 cm, 75°)

Draw $BC = 5$ cm. This is the side between the two given angles.

At $B$ , construct $\angle B = 45°$ . At $C$ , construct $\angle C = 75°$ . The two rays from $B$ and $C$ will meet at a point — call it $A$ .

The third angle $\angle A = 180° - 45° - 75° = 60°$ . You can verify with a protractor as a cross-check.

Why This Works

Each construction method uses a uniqueness condition for triangles:

SSS: Three sides fix the triangle completely — there is only one triangle possible (up to congruence).
SAS: Two sides and their included angle give a unique triangle because the third vertex is forced to one position.
ASA: Two angles fix the third angle ( $180°$ rule), and the included side fixes the size. Again, unique triangle.

The compass-and-ruler approach works because a compass draws circles (all points at a fixed distance), and the intersection of two circles or a circle and a line gives exact point locations.

Alternative Method

For SSS, you can also use a protractor-based approach: calculate an angle using the cosine rule ( $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$ ), then use SAS construction. But the compass method is faster and expected in CBSE board exams.

In board exams, always leave your construction arcs visible — do not erase them. Marks are given for showing the construction process, not just the final triangle.

Common Mistake

The biggest error in SSS construction: students set the compass to the wrong radius for the second arc. If sides are 5, 4, and 3 cm with base 5 cm, the arcs from the two endpoints must be 4 cm and 3 cm respectively — not both the same. Double-check which side goes with which endpoint before drawing the arc.