Quadrilateral
A quadrilateral refers to a four-sided polygon that has four angles. There are many types of quadrilaterals. The five most common types of quadrilaterals are the parallelogram, the rectangle, the square, the trapezoid, and the rhombus.
To determine geometric designs four important tools of geometry are-compass, straightedge, protractor, and ruler are usually used.
- Compass– helps to draw the different angles.
- Straightedge – it has two set squares, used to draw symmetric lines.
- Protractor – for determining and drawing correct and accurate angles.
- Ruler – Used to measure and draw lines to construct the figures.
Construction of Quadrilaterals
The following types of quadrilateral will be constructed:
- If four sides and one diagonal of a quadrilateral are given.
- If two diagonals and three sides of a quadrilateral are given.
- If two adjacent sides and three angles of a quadrilateral are given.
- If three sides and two included angles of a quadrilateral are given.
- If other special properties of a quadrilateral are known.
If four sides and one diagonal of a quadrilateral are given:
Follow the following steps to draw the quadrilateral:
Construct a quadrilateral PQRS where PQ= 5 cm, QR = 7 cm, RS = 6 cm, PS = 6.5 cm and PR = 8 cm.
Construct a rough sketch for reference first. Then follow the following:
Step 1: As we can see in the rough sketch, it is easy to see that ∆PQR can be constructed using SSS construction conditions. Draw ∆PQR.
Step 2: After this, we have to locate the fourth point S. This is on the side opposite to Q with reference to PR. For that, we have two measurements.
S is 6 cm away from point P. So, with P as centre an arc is drawn of radius 6 cm.
Step 3: S is 6.5 cm away from R. So with R as centre, draw an arc of radius 6.5cm
Step 4: S lies on the point of intersection of two arcs. Mark S and complete PQRS. PQRS is the required quadrilateral.
If two diagonals and three sides of a quadrilateral are given:
Follow the following steps to draw the quadrilateral:
Construct a quadrilateral ABCD, given that BC = 5.5 cm, AD = 6.5 cm, CD = 5 cm the diagonal AC = 6 cm and diagonal BD = 8 cm.
First, draw a rough sketch of the quadrilateral ABCD.
Step 1: Draw ∆ACD using SSS construction.
Step 2: With D as centre we draw an arc of radius 7 cm. With C as centre we draw an arc of radius 4.5 cm.
Step 3: With C as centre we draw an arc of radius 4.5 cm.
Step 4: Since B lies on both the arcs, B is the point intersection of the two arcs. ABCD is the required quadrilateral.
If two adjacent sides and three angles of a quadrilateral are given:
Follow the following steps to draw the quadrilateral:
Construct a quadrilateral MIST where MI = 3.5 cm, IS = 6.5 cm, ∠M = 75°, ∠I = 105° and ∠S = 120°.
Draw a rough sketch first.
Step 1: We first locate the points.
Step 2: Construct ∠IST = 120° at S
Step 3: Construct ∠IMT = 75° at M.
Step 4: We get the required quadrilateral MIST.
If three sides and two included angles of a quadrilateral are given:
Follow the following steps to draw the quadrilateral:
Construct a quadrilateral ABCD, where ST = 5 cm, TE = 6 cm, EP = 7.5 cm and ∠T = 105° and ∠E = 80°.
A rough sketch is drawn first
Step 1: Start with taking TE = 6 cm on Draw an angle of 105° along with TX.
Locate a point 5 cm away on this. We now have T, E, and S
Step 2: The fourth point P is on EY which is inclined at 80° to TE. So make ∠TEY = 80° at C on TE.
Step 3: D is at a distance of 7.5 cm on EY. With E as centre, draw an arc of length 7.5 cm. It cuts EY at P.
Step 4: Complete the quadrilateral STEP.
Step 5: STEP is the required quadrilateral.
Question 1: Draw a square with side 6cm.
Answer:
We can see that only the side is given but we know that square is a special figure with all the angles equal to right angle. We can use this information and draw the square.
Step 1: Draw a rough figure.
Step 2: Taking 6cm as base draw a line AB.
Step 3: Draw right angles at vertex A and B.
Step 4: Cut 6cm on the rays drawn from A and B respectively.
Step 5: Name the cut points C and D respectively.
Step 6: Join CD.
Step 7: ABCD is the required square.
Question 2: Construct a rhombus ABCD where AC = 6 cm and BD = 7 cm.
Answer:
The measurement of two diagonals is given. As we know,
The diagonals of a rhombus are perpendicular bisectors of each other.
Step 1: Firstly we will draw AC= 6cm and then we will draw its perpendicular bisector.
Step 2: Let them meet at 0.
Step 3: Cut off 3 cm lengths on either side of the drawn Bisector.
Step 4: We get B and D.
Step 5: Join the lines and we get the required rhombus ABCD.
Construct quadrilaterals when four sides and one diagonal is given.
Construct a quadrilateral PQRS where PQ = 4 cm, QR = 6 cm, RS = 5 cm, PS = 5.5 cm and PR= 7 cm
- Draw Δ PQR using SSS construction condition.
- With P as the centre, draw an arc of radius 5.5 cm.
- With R as the centre, draw an arc of radius 5 cm.
- S is the point of intersection of the two arcs. Also, mark S and complete PQRS.
- PQRS is the required quadrilateral.
Construct a quadrilateral ABCD where BC = 4.5 cm, AD = 5.5 cm, CD = 5cm and the diagonal AC = 5.5 cm, diagonal BD = 7 cm
- Draw ΔACD using SSS construction condition
- Taking D as the centre, draw an arc of radius 7 cm.
- Now let C be the centre, draw an arc of radius 4.5 cm
- Since B lies on both the arcs, B is the point intersection of the two arcs.
- Mark B and complete ABCD.
- ABCD is the required quadrilateral.
Construct a parallelogram when two consecutive sides and the included angle are given.
Construct a parallelogram ABCD with sides AB = 4 cm and AD = 5 cm and ∠A = 60
- First, construct a line segment AB = 4 cm and construct a 60 angle at point A.
- Now construct a line segment AD = 5 cm on the other arm of the angle. Then, place the sharp point of the compasses at B and make an arc 5 cm above B.
- Stretch your compasses to 4 cm, place the sharp end at D and draw an arc to intersect the arc drawn in step 2.
- Label the intersecting point C. Join C to D and B to C to form the parallelogram ABCD.
Construct a parallelogram when two consecutive sides and a diagonal are given.
Construct a parallelogram ABCD in which AB = 6cm, BC = 4.5cm and diagonal AC = 6.8 cm.
- Draw AB = 6 cm.
- With A as centre and radius 6.8 cm, draw an arc.
- With B as centre and radius, 4.5 cm draw another arc, cutting the previous arc at C.
- Join BC and AC.
- With A as centre and radius 4.5 cm, draw an arc.
- With C as centre and radius, 6 cm draw another arc, cutting the previously drawn arc at D.
- Join DA and DC
- ABCD is the required parallelogram.