Question

Construct a quadrilateral ABCD in which AB = 6 cm, BC = 4.2 cm, CD = 4.8 cm, DA = 4.2 cm and AC = 6.6 cm.

Answer 1

Given:

AB = 6 cm

BC = 4.2 cm

CD = 4.8 cm

DA = 4.2 cm

And AC = 6.6 cm

Step-wise calculation:

1. Existence check triangle inequalities.

For ΔABC:

6 + 4.2 > 6.6, 

6 + 6.6 > 4.2,

4.2 + 6.6 > 6 all true. 

So, ΔABC is possible.

For ΔADC:

4.2 + 4.8 > 6.6,

4.2 + 6.6 > 4.8,

4.8 + 6.6 > 4.2 all true. 

So, ΔADC is possible.

Thus, constructing ΔABC and ΔADC on the same base AC is feasible.

2. Compass-and-straightedge construction steps.

Step 1: Draw AC = 6.6 cm.

Step 2: Construct B:

With centre A and radius AB = 6 cm, draw an arc.

With centre C and radius BC = 4.2 cm, draw an arc.

Let one intersection of these two arcs on one side of AC be B.

Step 3: Construct D:

With centre A and radius AD = 4.2 cm, draw an arc.

With centre C and radius CD = 4.8 cm, draw an arc.

Let an intersection of these arcs on the opposite side of AC from B be D. This yields a convex quadrilateral.

Step 4: Join AB, BC, CD and DA to complete the quadrilateral ABCD.

This is the standard method for “four sides and one diagonal given”: build ΔABC first, then locate D by arcs from A and C, then join.

A quadrilateral ABCD with the given measurements can be constructed by first drawing AC = 6.6 cm, locating B as the intersection of arcs with radii 6 cm from A and 4.2 cm from C, locating D as the intersection of arcs with radii 4.2 cm from A and 4.8 cm from C on the opposite side of AC and joining the vertices AB, BC, CD, DA to obtain the required quadrilateral.