Question 7
Using the diagonal properties, construct a parallelogram whose diagonals are of lengths 7 cm and 5 cm, and intersect at an angle of 140°.
A parallelogram has diagonals that bisect each other, meaning they cut each other into two equal halves at their meeting point.
Step 1 — Draw the first diagonal
Let us start by drawing one of the diagonals. Draw a line segment AC with a length of 7 cm. Find the exact middle point of AC and label it O. So, the length of AO is half of AC.
The length of OC will also be 3.5 cm.
Step 2 — Draw the second diagonal
The second diagonal has a length of 5 cm. Since diagonals bisect each other, half of this diagonal will be 2.5 cm. Let this diagonal be BD, so BO = OD = 2.5 cm. At point O, use a protractor to draw a line segment that makes an angle of 140° with AC. Let's name this new line segment XOY, where X and Y are points on the line. Measure 2.5 cm from O along OX and mark point B. Measure 2.5 cm from O along OY (the opposite direction from OX) and mark point D. Now, BD is the second diagonal, and its total length is .
Step 3 — Complete the parallelogram
We now have the four corner points (vertices) of the parallelogram: A, B, C, and D. Connect these points with straight lines in order to form the sides of the parallelogram. Draw a line segment from A to B. Draw a line segment from B to C. Draw a line segment from C to D. Draw a line segment from D to A. The figure ABCD is the parallelogram we needed to construct.
Answer
The construction steps for the parallelogram are:
(i) Draw a line segment AC of length 7 cm. (ii) Mark the midpoint of AC as O, so AO = OC = 3.5 cm. (iii) At point O, draw a line segment BD such that BO = OD = 2.5 cm, and the angle is 140°. (iv) Join points A, B, C, and D in order (AB, BC, CD, DA) to form the parallelogram ABCD.
More questions in FIO
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(i) 30° (ii) 40° (iii) 90° (iv) 140°
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We saw that one of the properties of a rectangle is that its opposite sides are parallel. Can this be chosen as a definition of a rectangle? In other words, is every quadrilateral that has opposite sides parallel and equal, a rectangle?
Find the remaining angles in the following quadrilaterals.
Using the diagonal properties, construct a parallelogram whose diagonals are of lengths 7 cm and 5 cm, and intersect at an angle of 140°.
Using the diagonal properties, construct a rhombus whose diagonals are of lengths 4 cm and 5 cm.
Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides 4 cm.
Construct a kite whose diagonals are of lengths 6 cm and 8 cm.
Find the remaining angles in the following trapeziums—
Draw a Venn diagram showing the set of parallelograms, kites, rhombuses, rectangles, and squares. Then, answer the following questions—
(i) What is the quadrilateral that is both a kite and a parallelogram? (ii) Can there be a quadrilateral that is both a kite and a rectangle? (iii) Is every kite a rhombus? If not, what is the correct relationship between these two types of quadrilaterals?
If PAIR and RODS are two rectangles, find .
Construct a square with diagonal 6 cm without using a protractor.
CASE is a square. The points U, V, W and X are the midpoints of the sides of the square. What type of quadrilateral is UVWX? Find this by using geometric reasoning, as well as by construction and measurement. Find other ways of constructing a square within a square such that the vertices of the inner square lie on the sides of the outer square, as shown in Figure (b).
If a quadrilateral has four equal sides and one angle of 90°, will it be a square? Find the answer using geometric reasoning as well as by construction and measurement.
What type of a quadrilateral is one in which the opposite sides are equal? Justify your answer.
Hint: Draw a diagonal and check for congruent triangles.
Will the sum of the angles in a quadrilateral such as the following one also be 360°? Find the answer using geometric reasoning as well as by constructing this figure and measuring.
State whether the following statements are true or false. Justify your answers.
(i) A quadrilateral whose diagonals are equal and bisect each other must be a square.
(ii) A quadrilateral having three right angles must be a rectangle.
(iii) A quadrilateral whose diagonals bisect each other must be a parallelogram.
(iv) A quadrilateral whose diagonals are perpendicular to each other must be a rhombus.
(v) A quadrilateral in which the opposite angles are equal must be a parallelogram.
(vi) A quadrilateral in which all the angles are equal is a rectangle.
(vii) Isosceles trapeziums are parallelograms.