Quadrilaterals | FIO

Question 7

Using the diagonal properties, construct a parallelogram whose diagonals are of lengths $7\text{ cm}$ and $5\text{ cm}$ , and intersect at an angle of $140^\circ$ .

Solution

Understand the Question

In a parallelogram, the diagonals bisect each other.
Given two diagonals of lengths $7\text{ cm}$ $7 cm$ and $5\text{ cm}$ $5 cm$ intersecting at an angle of $140^\circ$ $14 0^{\circ}$ :
- Let $AC = 7\text{ cm}$ and $BD = 5\text{ cm}$ intersect at point $O$ .
- $O$ is the midpoint of both diagonals, so $AO = OC = \dfrac{7}{2} = 3.5\text{ cm}$ and $BO = OD = \dfrac{5}{2} = 2.5\text{ cm}$ .
By constructing these diagonals at the given angle and connecting their endpoints, the parallelogram $ABCD$ is formed.

Step 1 · Draw the First Diagonal

Draw a line segment $AC = 7\text{ cm}$ and mark its midpoint $O$ . Diagram 1

\begin{aligned} AO &= \dfrac{AC}{2} \\[0.6em] &= \dfrac{7\text{ cm}}{2} \\[0.6em] &= 3.5\text{ cm} \end{aligned}

Thus, $AO = OC = 3.5\text{ cm}$ .

Step 2 · Draw the Second Diagonal at $140^\circ$

At point $O$ , draw a straight line $XOY$ making an angle of $140^\circ$ with $AC$ . Diagram 2

Along this line, mark points $B$ and $D$ on opposite sides of $O$ such that:

\begin{aligned} BO = OD &= \dfrac{BD}{2} \\[0.6em] &= \dfrac{5\text{ cm}}{2} \\[0.6em] &= 2.5\text{ cm} \end{aligned}

The total length of the diagonal is $BD = 2.5\text{ cm} + 2.5\text{ cm} = 5\text{ cm}$ .

Step 3 · Join Vertices to Complete Parallelogram

Join the points $A$ to $B$ , $B$ to $C$ , $C$ to $D$ , and $D$ to $A$ in order. $ABCD$ is the required parallelogram.

Answer

The parallelogram $ABCD$ is constructed with diagonals of length $7\text{ cm}$ and $5\text{ cm}$ intersecting at $140^\circ$ .

Common Mistakes

Bisecting Diagonals Error: Forgetting to divide the diagonal lengths by $2$ ( $3.5\text{ cm}$ and $2.5\text{ cm}$ ) when marking distances from the intersection point $O$ .
Angle Placement: Measuring the $140^\circ$ angle from one of the end vertices rather than at the central intersection point $O$ .

More questions in FIO

Find all the other angles inside the following rectangles.

Draw a quadrilateral whose diagonals have equal lengths of $8\text{ cm}$ that bisect each other, and intersect at an angle of

(i) $30^\circ$

(ii) $40^\circ$

(iii) $90^\circ$

(iv) $140^\circ$

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We have seen how to get $90^\circ$ using paper folding. Now, suppose we do not have any paper but two sticks of equal length, and a thread. How do we make an exact $90^\circ$ using these?

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 $7\text{ cm}$ and $5\text{ cm}$ , and intersect at an angle of $140^\circ$ .

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 $4\text{ cm}$ .

Q10

Construct a kite whose diagonals are of lengths $6\text{ cm}$ and $8\text{ cm}$ .

Q11

Find the remaining angles in the following trapeziums—

Q12

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?

Q13

If PAIR and RODS are two rectangles, find $\angle \text{IOD}$ .

Q14

Construct a square with diagonal $6\text{ cm}$ without using a protractor.

Q15

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).

Q16

If a quadrilateral has four equal sides and one angle of $90^\circ$ , will it be a square? Find the answer using geometric reasoning as well as by construction and measurement.

Q17

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.

Q18

Will the sum of the angles in a quadrilateral such as the following one also be $360^\circ$ ? Find the answer using geometric reasoning as well as by constructing this figure and measuring.

Q19

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.