Quadrilaterals | A

Question 2

Can we complete this quadrilateral so that all its sides are of the same length?

Mark a point C whose distance from B and D is equal to AB (or AD). To do this, measure AB using a compass. Keeping this length as the radius, cut arcs from B and D.

Question diagram 1
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Solution

We can complete the quadrilateral to form a rhombus, which has all its sides of the same length.

Step 1 — Understand the properties

A rhombus is a special type of quadrilateral. All four sides of a rhombus are equal in length. We are given two sides, AB and AD, and the angle at A. For the quadrilateral to have all sides of the same length, we must have AB=BC=CD=DAAB = BC = CD = DA.

Diagram 1

Step 2 — Measure the side length

Let us use a compass to measure the length of side AB. We place the compass needle at point A and the pencil tip at point B. This sets the radius of our compass to the length of AB.

Diagram 2

Step 3 — Locate point C

Now, we need to find point C. Its distance from B must be equal to AB. Its distance from D must also be equal to AB (because ADAD is equal to ABAB for a rhombus). Keeping the compass opening (radius) equal to AB, we place the compass needle at point B. We draw an arc. Then, we place the compass needle at point D. We draw another arc. The point where these two arcs intersect is our point C.

Diagram 3

Step 4 — Complete the quadrilateral

Finally, we use a ruler to draw straight lines. We connect point B to point C. We also connect point D to point C. This completes the quadrilateral ABCD. Since we constructed it such that AB=ADAB = AD (as shown in the diagram) and BC=ABBC = AB and CD=ABCD = AB, all four sides are equal. Therefore, ABCD is a rhombus.

Diagram 4

Answer

(i) Yes, we can complete this quadrilateral so that all its sides are of the same length. (ii) The completed quadrilateral is a rhombus.

More questions in A

Q1

Are squares the only quadrilaterals that have equal sidelengths? Let us explore this question through construction.

Draw two equal sides AD and AB, that are not perpendicular to each other.

Q2

Can we complete this quadrilateral so that all its sides are of the same length?

Mark a point C whose distance from B and D is equal to AB (or AD). To do this, measure AB using a compass. Keeping this length as the radius, cut arcs from B and D.

Q3

Place two rubber bands perpendicular to each other, forming diagonals of equal length. Join the ends.

What is the quadrilateral that you get? Justify your answer.

Q4

Extend one of the diagonals on both sides by 2 cm.

What quadrilateral will you get now? Justify your answer.

Q5

Take two cardboard cutouts of an equilateral triangle of sidelength 8 cm.

  • Can you join them to get a quadrilateral?
  • What type of a quadrilateral is this? Justify your answer.
Q6

Take two cardboard cutouts of an isosceles triangle with sidelengths 8 cm, 8 cm, and 6 cm.

  • What are the different ways they can be joined to get a quadrilateral?
  • What quadrilaterals are these? Justify your answers.
Q7

Take two cardboard cutouts of a scalene triangle with sides 6 cm, 9 cm, and 12 cm.

  • What are the different ways they can be joined to get a quadrilateral?
  • Are you able to identify the different quadrilaterals that are obtained by joining the triangles? Justify your answer whenever you identify a quadrilateral.
Q8

Which Quad?

Gameplay

  1. Fold a sheet into half.
  2. Now, fold it once more into a quarter.
  3. Make a triangular crease at the corner that is at the middle of the paper.
  4. Open the sheet. What is the shape formed by the creases?
  5. How would you fold the quarter paper to get the kinds of creases shown in the following image.
  6. How would you fold the quarter paper such that a square is formed?
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