Grade 8 Construction of Quadrilaterals

Definition: A **quadrilateral** is a closed two-dimensional shape with four straight sides and four vertices (corners).

Elements: A quadrilateral has:

• 4 Sides
• 4 Angles
• 4 Vertices
• 2 Diagonals (line segments connecting opposite vertices)

Note: The sum of the interior angles of any quadrilateral is 360 degrees.

Step 1: Draw a rough sketch of the quadrilateral and label the vertices and given measurements. This helps visualize the construction.

Step 2: Draw the diagonal AC = 6 cm as the base.

A------------------C
      (6 cm)
                                    

Step 3: With A as the center, draw an arc with radius AB = 4 cm above AC.

Step 4: With C as the center, draw an arc with radius BC = 5 cm above AC. This arc should intersect the previous arc. The intersection point is vertex B.

Step 5: Join AB and BC. You have constructed triangle ABC.

        B
       / \
      /   \
     /     \
    /       \
A------------C
     

Step 6: Now, we need to locate vertex D. With A as the center, draw an arc with radius AD = 3.5 cm below AC.

Step 7: With C as the center, draw an arc with radius CD = 4.5 cm below AC. This arc should intersect the arc drawn from A. The intersection point is vertex D.

Step 8: Join AD and CD.

        B
       / \
      /   \
     /     \
    /       \
A------------C
    \       /
     \     /
      \   /
       \ /
        D
     

Result: ABCD is the required quadrilateral.

Step 1: Draw a rough sketch. Notice we have two diagonals and three sides. We can start by constructing a triangle involving one diagonal and two sides connected to its endpoints.

Step 2: Draw diagonal PR = 6 cm as the base.

P------------------R
      (6 cm)
                                    

Step 3: With P as the center, draw an arc with radius PQ = 4 cm above PR.

Step 4: We also know the length of diagonal QS = 7 cm. Point S is 7 cm away from Q. Point Q is on the arc drawn from P. This approach is tricky. Let's rethink the base.

Alternative Approach:

Consider triangle PQR. We know PQ = 4, QR = 5, and PR = 6. We can construct triangle PQR first.

Step 2 (Revised): Draw PQ = 4 cm.

P------------Q
   (4 cm)
                                    

Step 3 (Revised): With P as center, draw an arc with radius PR = 6 cm. With Q as center, draw an arc with radius QR = 5 cm. These arcs intersect at R. Join PR and QR. You have triangle PQR.

          R
         / \
        /   \
       /     \
      /       \
P------------Q
     

Step 4 (Revised): Now locate S. We know RS = 5.5 cm and QS = 7 cm.

Step 5 (Revised): With R as center, draw an arc with radius RS = 5.5 cm.

Step 6 (Revised): With Q as center, draw an arc with radius QS = 7 cm. This arc intersects the arc from R at point S.

Step 7 (Revised): Join PS and RS.

          R
         / \
        /   \
       /     \
      /       \
P------------Q
 \           /
  \         /
   \       /
    \     /
     \   /
      \ /
       S
     

Result: PQRS is the required quadrilateral.

Step 1: Draw a rough sketch. We have two consecutive angles and the three sides involved in those angles.

Step 2: Draw segment BC = 4.5 cm as the base.

B------------C
  (4.5 cm)
                                    

Step 3: At point B, use a protractor to draw a ray BX making an angle of 60 degrees with BC.

    X
   /
  /
 /
B------------C
     

Step 4: With B as the center, draw an arc with radius AB = 5 cm intersecting the ray BX. The intersection point is vertex A.

    A (5 cm)
   / \
  /   \
 /     \
B------------C
     

Step 5: At point C, use a protractor to draw a ray CY making an angle of 75 degrees with BC (on the same side of BC as A).

    A
   / \
  /   \
 /     \
B------------C----Y
           /
          / (75 deg)
         /
     

Step 6: With C as the center, draw an arc with radius CD = 4 cm intersecting the ray CY. The intersection point is vertex D.

    A
   / \
  /   \
 /     \
B------------C
 \           /
  \         / (4 cm)
   \       /
    D
     

Step 7: Join AD.

    A
   / \
  /   \
 /     \
B------------C
 \           /
  \         /
   \       /
    D
     

Result: ABCD is the required quadrilateral.

Property: A rhombus is a quadrilateral with all four sides equal in length. Opposite angles are equal.

Step 1: Draw a side, say AB = 5 cm.

A------------B
   (5 cm)
     

Step 2: At point A, draw a ray AX making an angle of 60 degrees (the given angle).

    X
   /
  /
 /
A------------B
     

Step 3: With A as the center, draw an arc with radius AD = 5 cm (since all sides are 5 cm) intersecting the ray AX. The intersection is vertex D.

    D (5 cm)
   / \
  /   \
 /     \
A------------B
     

Step 4: Now locate vertex C. With D as the center, draw an arc with radius DC = 5 cm.

Step 5: With B as the center, draw an arc with radius BC = 5 cm. This arc intersects the arc from D at vertex C.

    D --------- C (5 cm)
   / \         /
  /   \       /
 /     \     / (5 cm)
A------------B
     

Step 6: Join DC and BC.

    D --------- C
   /           /
  /           /
 /           /
A------------B
     

Result: ABCD is the required rhombus.