Quadrilaterals | FIO

Question 9

Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides 4 cm.

Solution

We can form a quadrilateral by joining two equilateral triangles along one of their sides.

Step 1 — Understanding Equilateral Triangles

Let us start with what we know about an equilateral triangle. An equilateral triangle is a triangle where all three sides are equal in length. Also, all three angles inside an equilateral triangle are equal.

We are given two equilateral triangles. Each has sides of 4 cm.

Since all angles in an equilateral triangle are equal, and the sum of angles in any triangle is $180^\circ$ :

$\text{Each angle} = \frac{180^\circ}{3}$

$= 60^\circ$

$\boxed{\text{Each angle of an equilateral triangle} = 60^\circ}$

Diagram 1

Step 2 — Finding the Sides of the Quadrilateral

Let us imagine we join two such equilateral triangles, say $\triangle ABC$ and $\triangle ADC$ , along their common side $AC$ . This creates a new shape, a quadrilateral $ABCD$ .

From $\triangle ABC$ , we know its sides are: $AB = 4 \text{ cm}$ $BC = 4 \text{ cm}$ $AC = 4 \text{ cm}$

From $\triangle ADC$ , we know its sides are: $AD = 4 \text{ cm}$ $CD = 4 \text{ cm}$ $AC = 4 \text{ cm}$

The quadrilateral $ABCD$ has four outer sides: $AB$ , $BC$ , $CD$ , and $DA$ . Let us list their lengths: $AB = 4 \text{ cm}$ $BC = 4 \text{ cm}$ $CD = 4 \text{ cm}$ $DA = 4 \text{ cm}$

$\boxed{\text{All sides of the quadrilateral ABCD are 4 cm}}$

Step 3 — Finding the Angles of the Quadrilateral

Now, let us find the angles of the quadrilateral $ABCD$ . The angles of the quadrilateral are $\angle A$ , $\angle B$ , $\angle C$ , and $\angle D$ .

Angle $A$ of the quadrilateral is $\angle DAB$ . This angle is formed by combining $\angle BAC$ from $\triangle ABC$ and $\angle DAC$ from $\triangle ADC$ . $\angle DAB = \angle BAC + \angle DAC$ $= 60^\circ + 60^\circ$ $= 120^\circ$

Angle $C$ of the quadrilateral is $\angle BCD$ . This angle is formed by combining $\angle BCA$ from $\triangle ABC$ and $\angle DCA$ from $\triangle ADC$ . $\angle BCD = \angle BCA + \angle DCA$ $= 60^\circ + 60^\circ$ $= 120^\circ$

Angle $B$ of the quadrilateral is $\angle ABC$ . This is one of the angles of $\triangle ABC$ . $\angle ABC = 60^\circ$

Angle $D$ of the quadrilateral is $\angle ADC$ . This is one of the angles of $\triangle ADC$ . $\angle ADC = 60^\circ$

Answer

(i) The sides of the quadrilateral ABCD are 4 cm each. (ii) The angles of the quadrilateral ABCD are angle A = 120 degrees, angle B = 60 degrees, angle C = 120 degrees and angle D = 60 degrees.

More questions in FIO

Find all the other angles inside the following rectangles.

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

(i) 30° (ii) 40° (iii) 90° (iv) 140°

Consider a circle with centre O. Line segments PL and AM are two perpendicular diameters of the circle. What is the figure APML? Reason and/or experiment to figure this out.

We have seen how to get 90° 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° 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 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.

Q10

Construct a kite whose diagonals are of lengths 6 cm and 8 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 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°, 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°? 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.