Is there a 4-sided figure in which all the sides are equal in length but is not a square? If such a figure exists, can you construct it?
Yes, a rhombus is a 4-sided figure where all sides are equal but it is not a square.
In a rhombus, all four sides are equal in length, but the angles are not 90°.
A square is a special case of a rhombus where all angles are right angles.
Construction of a rhombus with 5 cm sides:
Step 1: Draw AB = 5 cm.
Step 2: Choose an angle less than 90° (e.g., 60°) at A.
Step 3: Draw AD = 5 cm at that angle.
Step 4: With B as centre, draw an arc of radius 5 cm.
Step 5: With D as centre, draw an arc of radius 5 cm.
Step 6: The intersection gives point C.
Step 7: Join BC and CD.
Conclusion: The resulting figure ABCD is a rhombus with all sides 5 cm but angles not equal to 90°.
Why it works: Point C lies at the intersection of two arcs of radius 5 cm from B and D, so BC = 5 cm and DC = 5 cm. Since AB = AD = 5 cm by construction, all four sides are equal, forming a rhombus with non-right angles.
Explanation
The chapter hints that this construction uses the same technique as the House problem - finding a point equidistant from two given points. Point D must be 5 cm from both B and C. The intersection of two circles/arcs with centres B and C gives point D.
Solution Steps
Step 1: Draw AB = 5 cm.
Step 2: At A, draw AD = 5 cm at an angle (e.g., 60°).
Step 3: With B as centre, draw arc of radius 5 cm.
Step 4: With D as centre, draw arc of radius 5 cm.
Step 5: Mark intersection as C.
Step 6: Join BC and CD.
Step 7: ABCD is a rhombus with all sides equal but angles not 90°.