Oops! It appears that you have disabled your Javascript. In order for you to see this page as it is meant to appear, we ask that you please re-enable your Javascript!

Practical Geometry - Class 8 : Notes


Constructing a Quadrilateral:

1. Steps to construct a quadrilateral when lengths of four sides and a diagonal are given
Let us understand it by an example:

Example 1: Construct a quadrilateral ABCD such that AB = 5 cm, BC = 7 cm, CD = 6 cm, DA = 6.5 cm and AC = 8 cm.
Solution:
Step 1: Construct ΔABC using SSS criterion.Step 2: Now, to draw the fourth point D. Keep A as center and draw and arc of radius 6.5 cm on the opposite side of point B.Step 3: Draw an another arc keeping center as C of radius 6 cm, such that it intersects the arc drawn in above step.Step 4: The intersecting point of both the arcs will be point D. Draw the lines from point A and C to complete the quadrilateral ABCD.

Example 2: Construct a parallelogram MORE with OR = 6cm, RE = 4.5 cm and EO = 7.5cm.
Solution:
Step 1: Construct ΔEOR using SSS criterionStep 2: To draw point M, take O and E as centers with arcs of radius 4.5 cm and 6 cm respectively. These two must intersect and this is the point M.Step 3: To complete the parallelogram MORE, join O and E to M.

2. Steps to construct a quadrilateral when two diagonals and three sides are given
Let us understand it by an example:

Example 1: Construct a quadrilateral ABCD such that the diagonals AC = 6.5 cm and BD = 8 cm. The measure of the other sides will be BC = 5.5 cm, AD = 6.5 cm and CD = 6 cm.
Solution:
Step 1: Construct ΔACD using SSS criterion.Step 2: Now, to draw the fourth point B. Keep D as center and draw and arc of radius 8 cm on the opposite side of point D.Step 3: Draw an another arc keeping center as C of radius 5.5 cm, such that it intersects the arc drawn in above step.Step 4: The intersecting point of both the arcs will be point B. Draw the lines from point A and C to complete the quadrilateral ABCD.Example 2: Draw the rhombus BEND such that BN = 5.6 cm and DE = 6.5 cm.
Solution:
Step 1: Draw a line segment BN of length 6.5 cm. Now, draw perpendicular bisectors of BN with the intersection point as O.Step 2: Take O as center and draw arcs of length 3.25 cm which intersects the perpendicular bisector at point D and E.Step 3: To complete the rhombus BEND, join D to B and E to N.

3. Steps to construct a quadrilateral when two adjacent sides and three angles are given
Let us understand it by an example:

Example 1: Construct a quadrilateral ABCD such that the AB = 4.5 cm and BC = 7.5 cm. The measure of the angles are ∠A = 75ᵒ, ∠B = 105ᵒ and ∠C = 120ᵒ.
Solution:
Step 1: Draw AB = 4.5 cm. Then make ∠B = 105ᵒ and draw BC = 7.5 cm.Step 2: Make ∠C = 120ᵒ.Step 3: Make ∠A = 75ᵒ such that it intersects the line extended from point C. At last, complete the quadrilateral ABCD.Example 2: Construct a quadrilateral PLAN such that PL = 4 cm, LA = 6.5 cm, ∠P = 90ᵒ, ∠A = 110ᵒ and ∠N = 85ᵒ.
Solution:We know that sum of angles of a quadrilateral is 360ᵒ.
So, ∠P + ∠L + ∠A + ∠N = 360ᵒ.
90ᵒ + ∠L + 110ᵒ + 85ᵒ = 360ᵒ
Thus, ∠L = 75ᵒ
Step 1: Draw line segment PL of length 4 cm. Now, draw an angle of 75ᵒ at point L; cut a line segment LA of 6.5 cm from point L.Step 2: From point A, draw an angle of 110ᵒ.Step 3: From point P, draw an angle of 90. This ray will meet the ray from point A intersecting at point N.

3. Steps to construct a quadrilateral when three sides and two angles are given
Let us understand it by an example:

Example 1: Construct a quadrilateral ABCD such that the AB = 5 cm, BC = 6 cm and CD = 7.5 cm. The measure of the angles are ∠B = 105ᵒ and ∠C = 80ᵒ.
Solution:
Step 1: Draw line BC = 6cm. Now, draw ∠CBA = 105ᵒ and AB = 5cm.Step 2: Draw ∠C = 80ᵒ towards point B.Step 3: Draw an arc of length 7.5 cm from point C, since, it is given CD = 7.5 cm.Step 4: Join the line AD and complete the quadrilateral ABCD.

Example 2: Draw a quadrilateral TRUE with TR = 3.5 cm, RU = 3 cm, UE = 4 cm, ∠R = 75ᵒ and ∠U = 120ᵒ.
Solution:
Step 1: Draw line segment RU of 3 cm. Now, make angle ∠U = 120ᵒ. Keeping U as centre cut a line segment UE of 4 cm.Step 2: Draw ∠R = 75ᵒ. And keeping R as centre cut a line segment RT of 3.5 cm.Step 3: To complete quadrilateral TRUE, join T to E.



1 Comment

Leave a Reply