Question 8
Using the diagonal properties, construct a rhombus whose diagonals are of lengths 4 cm and 5 cm.
A rhombus is a special type of quadrilateral where its diagonals cut each other in half and meet at a right angle.
Step 1 — Draw the first diagonal
Let us start by drawing one of the diagonals. We will draw a line segment AB with a length of 4 cm.
Step 2 — Find the midpoint
The diagonals of a rhombus bisect each other (cut each other into two equal halves). We need to find the midpoint of our first diagonal, AB. Let O be the midpoint of AB. The length of AO will be half of the total length of AB.
So, we mark a point O on AB such that AO is 2 cm long.
Step 3 — Draw the perpendicular bisector
Another important property of a rhombus is that its diagonals are perpendicular to each other (they meet at a 90-degree angle). We will draw a line that passes through O and is perpendicular to AB. This line will be where our second diagonal lies.
Step 4 — Mark the second diagonal
The second diagonal has a length of 5 cm. Since the diagonals bisect each other, half of this diagonal will be divided by 2.
On the perpendicular line we drew through O, we will mark two points, C and D. Point C will be 2.5 cm away from O in one direction. Point D will be 2.5 cm away from O in the opposite direction. So, OC = OD = 2.5 cm. The total length of the diagonal CD will be .
Step 5 — Complete the rhombus
Now we have the four corner points of our rhombus: A, B, C, and D. We will connect these points with straight lines to form the sides of the rhombus. Join AC, then CB, then BD, and finally DA. The figure ACBD is the rhombus we wanted to construct. Its diagonals AB and CD are 4 cm and 5 cm long, they bisect each other at O, and they are perpendicular.
Answer
(i) Draw a line segment AB of length 4 cm. (ii) Locate the midpoint O of AB, so that AO = OB = 2 cm. (iii) Draw a line perpendicular to AB passing through O. (iv) On this perpendicular line, mark points C and D such that OC = OD = 2.5 cm. (v) Join points A, C, B, and D in order (AC, CB, BD, DA) to form the rhombus ACBD.
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(vii) Isosceles trapeziums are parallelograms.