Quadrilaterals | FIO

Question 9

Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides $4\text{ cm}$ .

Solution

Understand the Question

An equilateral triangle has all 3 sides equal and all 3 interior angles equal to $\dfrac{180^\circ}{3} = 60^\circ$ .
When two identical equilateral triangles with side length $4\text{ cm}$ are joined along a common side, they form a rhombus (a quadrilateral with four equal sides).
The four outer boundary sides become the sides of the quadrilateral, each measuring $4\text{ cm}$ .
The angles at the shared base vertices combine ( $60^\circ + 60^\circ = 120^\circ$ ), while the other two opposite angles remain $60^\circ$ .

Step 1 · Find the Angles of an Equilateral Triangle

For any equilateral triangle, all three sides are equal and all three interior angles are equal. Diagram 1

\begin{aligned} \text{Each angle} &= \dfrac{180^\circ}{3} \\[0.6em] &= 60^\circ \end{aligned}

Step 2 · Find the Sides of the Quadrilateral

Let the two equilateral triangles be $\Delta \text{ABC}$ and $\Delta \text{ADC}$ , joined along the common side $\text{AC}$ to form quadrilateral $\text{ABCD}$ .

Sides of $\Delta \text{ABC}$ :

\begin{aligned} \text{AB} &= 4\text{ cm} \\ \text{BC} &= 4\text{ cm} \\ \text{AC} &= 4\text{ cm} \end{aligned}

Sides of $\Delta \text{ADC}$ :

\begin{aligned} \text{AD} &= 4\text{ cm} \\ \text{CD} &= 4\text{ cm} \\ \text{AC} &= 4\text{ cm} \end{aligned}

The four outer sides of quadrilateral $\text{ABCD}$ are:

\begin{aligned} \text{AB} &= 4\text{ cm} \\ \text{BC} &= 4\text{ cm} \\ \text{CD} &= 4\text{ cm} \\ \text{DA} &= 4\text{ cm} \end{aligned}

Step 3 · Find the Angles of the Quadrilateral

Combine the adjacent angles at vertices $\text{A}$ and $\text{C}$ :

\begin{aligned} \angle \text{DAB} &= \angle \text{BAC} + \angle \text{DAC} \\[0.6em] &= 60^\circ + 60^\circ \\[0.6em] &= 120^\circ \end{aligned}

\begin{aligned} \angle \text{BCD} &= \angle \text{BCA} + \angle \text{DCA} \\[0.6em] &= 60^\circ + 60^\circ \\[0.6em] &= 120^\circ \end{aligned}

The other two angles are the single vertex angles of the individual triangles: $\angle \text{ABC} = 60^\circ$ $\angle \text{ADC} = 60^\circ$

Answer

Sides: $4\text{ cm}, 4\text{ cm}, 4\text{ cm}, 4\text{ cm}$

Angles: $120^\circ, 60^\circ, 120^\circ, 60^\circ$

Common Mistakes

Including the Common Side: Counting the shared internal side $\text{AC}$ as one of the sides of the quadrilateral. A quadrilateral has only $4$ outer boundary sides.
Not Combining Adjacent Angles: Forgetting that vertices $\text{A}$ and $\text{C}$ combine two $60^\circ$ angles from both triangles to form $\angle \text{A} = 120^\circ$ and $\angle \text{C} = 120^\circ$ .

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.