NCERT Solutions for Class 8 Maths Chapter 4: Practical Geometry - Exercise 4.5

Exercise 4.5

Refer to page 23-28 for Exercise 4.5 in the PDF.

1. Draw the following:

The square READ with $RE = 5.1cm$

Ans: In the figure of square, all the sides are of the same measure and also all the interior angles are of $90^\circ $ measure. Therefore, the square READ can be drawn as follows: 

STEP 1 :Firstly, we will draw a rough sketch of square READ

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STEP 2 :Now, draw a line segment RE of length $5.1cm$ and an angle of $90^\circ $

at a point R and E.

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STEP 3: Now, as the vertex A and D are of length $5.1cm$ away from vertex E and R,

Cut the line segments EA and RD , each of length $5.1cm$ from these rays.

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STEP 4: Now, join point D to A.

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Here,  READ is the required square.

2. Draw the following:

A rhombus whose diagonals bisect are $5.2cm$ and $6.4cm$ long.

Ans: In the figure of rhombus , the diagonals bisect each other at $90^\circ $. Therefore, the given rhombus ABCD can be drawn as follows.

STEP 1:

Firstly, draw a rough sketch of this rhombus ABCD  as follows.

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STEP 2: Now, draw a line segment AC of length $5.2cm$ and draw its perpendicular bisector. And let it intersect the line segment AC at point O.

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STEP 3: Draw arcs of $\begin{gathered}

\frac{{6.4}}{2} = 3.2cm \hfill \\

\hfill \\ 

\end{gathered} $ on both sides of this perpendicular bisector. Let at point B and D, the arcs intersect the perpendicular bisector.

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STEP 4: Now, join points B and D with points A and C.

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Here, ABCD is the required rhombus.

3. Draw the following:

A rectangle with adjacent sides of length 5 cm and 4 cm.

Ans: In the figure of a rectangle, opposite sides have their lengths of the same measures and all the interior angles of a rectangle are of $90^\circ $ measure. The given rectangle ABCD can be drawn as follow:

STEP 1: Firstly, we will draw a rough sketch of a rectangle ABCD.

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STEP 2: Now, draw a line segment AB of 5 cm and an angle of $90^\circ $ at point A and B.

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STEP 3: Here,  As the vertex C and D are 4 cm away from vertex B and A respectively, cut the line segments AD and BC, each of 4 cm, from these rays.

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STEP 4: Join point D to C.

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ABCD is the required rectangle.

4. Draw the following : 

A parallelogram OKAY where $OK = 5.5cm$ and $KA = 4.2cm$.

Ans: In the figure of parallelogram, opposite sides are equal and parallel to each other.The given parallelogram OKAY can be drawn as follows.

STEP 1: Firstly, we will draw a rough sketch of this parallelogram OKAY.

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STEP 2: Now, draw a line segment OK of $5.5cm$ and a ray at point K at a convenient angle.

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STEP 3: Now, draw a ray at point O parallel to the ray at K. As the vertices A and Y are $4.2cm$ Away from the vertices K and O respectively, cut line segments KA and OY, each of  $4.2cm$ from these rays.

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STEP 4: Join point Y to A

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OKAY is the required rectangle.

Chapter 4 - Quadrilaterals

Chapter 4- Quadrilaterals covers the following topics.

• What is Quadrilateral?

• How to construct a Quadrilateral PQRS?

• What is Rhombus?

• How to construct a Rhombus?

• What is Parallelogram?

• How to construct a Parallelogram?

Question: What is Quadrilateral?

Ans: Quadrilateral is a four-sided polygon that has four vertices with four different angles.

The sum of all the angles of  Quadrilateral is equal 360*. This is also known as By sum property of Quadrilateral. It is of two types-

• Simple Quadrilateral

• .Compound Quadrilateral.

A quadrilateral can have two diagonals that can be drawn to the opposite side. Some examples of Quadrilateral are as follows

• Rhombus

• Parallelogram

• Square

• Rectangle. 

• Trapezium

A quadrilateral is always a closed shaped structure. If any quadrilateral ABCD has two adjacent parallel sides, then it can be said to be Trapezium. 

The sum of its all sides can determine the parameter of any Quadrilateral ABCD.

Question: How to construct a Quadrilateral PQRS

Ans:  Given-  MN = 4.5 cm, NO = 5 cm, OP = 6 cm, PM = 7 cm and       MO= 8 cm.

Step 1: Draw a line of MO=8 cm.

Step 2: Take M as a centre and draw an arc of 4.5cm,

Step 3: Take O as a centre and draw an arc of 5cm and meet the first arc.

Step 4: Now give the point of intersection of two arcs as N and join the points.

Hence, you will have a triangle MNO.

Step 5: Again, take M as a centre and draw an arc of 7cm

Step 6: Again, take O as a centre and draw an arc of 6cm.

Step 7: Now, name the point of intersection of two arcs as point P and join all points.

Finally, you will get the MNOP is a Quadrilateral with the diagonal MO.

Question: What is Rhombus?

Ans:   Rhombus is a quadrilateral whose all four sides have equal length. Each angle of Rhombus is equal to 90*. It has four vertices and can have two diagonals of the opposite side, which divides the Rhombus into two equilateral triangles.

The diagonals of the rhombus ABCD intersect each other at 90* angles. They also bisect each other. When two diagonals intersect each other, they form four triangles of equal areas. The diagonals of Rhombus ABCD bisects the opposite angles of the Rhombus. The city of Rhombus ABCD can be calculated by the formula- Product of Length and the width.

The parameter of Rhombus ABCD is the four times of its one side.

i.e., 4* side

The Rhombus ABCD is similar to a square as it has all four equal sides. But there is a difference between a square and the Rhombus; the Rhombus has opposite equal angles while the square has all four right corners.

Rhombus ABCD is also a type of parallelogram, but Rhombus has equal sides, but a parallelogram has only two equal opposite sides. Adjacent angles of the Rhombus ABCD are supplementary to each other, which means the Sum of two adjacent angles is equals to 180*.

The most common example of a Rhombus is a Diamond.

Question: How to construct a Rhombus?

Ans: Given-  PR= 8 cm and QS=10 cm

Construction:

Step 1: Draw a line segment PR= 8cm

Step 2: Take P as a centre and draw arcs of radius more than half of PR.

Step 3: Take R as a centre and draw arcs using the same radius which cuts the previous arcs and you will get two points

Step 4: Join them, and you will get a perpendicular bisector M to the line segment PR.

Step 5: Now, take O as a centre and draw the arcs with the radius half of the line segment QS.

Step 6:  You will get the points Q and S.

Step 7: Join the Points P, Q, R, S.

Hence, PQRS is your required rhombus.

Question: What is Parallelogram?

Ans: It is a Quadrilateral which has two Parallel and Equal opposite sides. The Diagonals of a parallelogram divided it into two equal triangles. The different angles of a parallelogram are always fair. The two diagonals of Parallelogram intersect each other at a point M.

Opposite sides of a parallelogram PQRS can never intersect with each other as they are parallel to each other. Any line which passes through the Midpoint M divides the area of Parallelogram PQRS. The opposite angles of any parallelogram always equal to 180*, that is different angles of Parallelogram are supplementary to each other. A parallelogram is also a polygon with the opposing parallel sides.

The two diagonals of Parallelogram PQRS bisect each other when they are drawn. The sum of all the angles of the Parallelogram is always equated to 360*.

The area of the Parallelogram is equalled to the half product of its base and height. 

The most common example of a Parallelogram includes tables etc.

Question: How to construct a Parallelogram PQRS?

Ans: Given-  PQ= 6cm   PS= 5cm  and an angle p =60*

Construction:

Step 1:  Draw the line segment PQ = 6cm.

Step 2:  Construct an angle P= 60*

Step 3:  Take P as a centre and draw an arc of 5 cm.

Step 4:  The arc meets the ray of angle P at point S.

Step 5:  Take Q as a centre and draw an arc of 5cm above Q.

Step 6:   Now take S as a centre and draw an arc of 6 cm, which cuts the Arc made by Point Q ( in step 5).

Step 7:  Now join all the points P, Q, R, S.

Hence PQRS is your required Parallelogram.

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