Constructing Shapes
Constructing quadrangles
A quadrangle with equal sides and right angles is a _________.
Let's draw a quadrangle! All sides should be four centimetres long, and all angles in the corners should be right angles, 90 degrees. Start by drawing one of the sides with a ruler - a line four centimetres long. Mark one endpoint A, and the other B. Now let's draw the next side.
It should form an angle of 90 degrees with the first line. We use a protractor to get the right angle. Place the straight part of the protractor along the line AB so that the angle mark ends at point A. Mark 90 degrees with a small dash. Then use the ruler and draw a line that has the angle 90 degrees to the line AB.
Measure four centimetres on the new line and mark that point with a D. Similarly, draw a third line, the third side. It should form an angle of 90 degrees with the line AB at point B. Measure the distance four centimeters on the new line and mark that point C. Now: Draw the last side.
A line between points C and D. Look, it's a square! A square is a quadrangle with equal sides and right angles. Now, we will draw a quadrangle with four different lengths, an irregular quadrangle. We call the corners A, B, C and D.
The side AB should be 4 centimetres, BC 7 centimetres, CD 5 centimetres and DA 5,5 [5.5] centimetres. We also know that the length from A to C is 8 centimetres. That is one of the diagonals to the quadrangle. Start by drawing this diagonal line that is 8 centimetres long. Mark one endpoint A and the other C.
Why C and not B? Because the second end point of the diagonal becomes the third corner of the quadrangle. Adjust the radius of a compass to the length of the first side in the quadrangle, four centimetres. Place the compass needle in point A and draw an arc above the diagonal AC. Now: Change the radius of the compass to the second side's length, seven centimetres.
Place the compass needle in point C. Draw an arc above the diagonal AC that crosses the first arc. Mark B where the arcs cross each other. Draw a line between points A and B and another line between C and B. Do you see that we have drawn a triangle?
All quadrangles consist of two triangles with a common side - the diagonal. If we can draw the two triangles - then we have drawn the quadrangle. Come along and we will draw the second triangle. Set the radius of the compass to the length of the third side, 5,5 [5.5] centimetres. Place the compass needle in point A, and draw an arc below the diagonal AC.
Change the radius of the compass to the length of the fourth side, five centimetres. Place the needle in point C and draw an arc under the diagonal AC - that crosses the previous arc. Mark D where the arcs cross. Now draw a line between A and D and another between C and D. The second triangle is ready, and the quadrangle is done too!
All quadrangles have: four sides, four angles, and two diagonals. There are 10 properties altogether. In the first example we found the length of four sides and the size of four angles. There were eight properties. In the second example, we found the length of four sides and one diagonal.
There were only five properties, but it was enough! If we have a ruler, compass, protractor and pen, we can draw any quadrangle if we know at least five of its properties.