Maharashtra Board Class 8 Maths part 1 Chapter 8 Quadrilateral Constructions and Types PDF Download

Quadrilateral: Constructions And Types

Let's Recall

Construct the triangles with given measures.

(1) Triangle ABC: l(AB) = 5 cm, l(BC) = 5.5 cm, l(AC) = 6 cm

(2) Triangle DEF: m angle D = 35°, m angle F = 100°, l(DF) = 4.8 cm

(3) Triangle MNP: l(MP) = 6.2 cm, l(NP) = 4.5 cm, m angle P = 75°

(4) Triangle XYZ: m angle Y = 90°, l(XY) = 4.2 cm, l(XZ) = 7 cm

Every quadrilateral has 4 angles, 4 sides and 2 diagonals. So there are 10 elements of each quadrilateral.

Let's Learn

Construction Of A Quadrilateral

We can construct a quadrilateral if we know the measures of some specific 5 elements out of 10. Constructions of triangles are the basis of constructions of quadrilaterals. This will be clear from the following examples.

(I) To Construct A Quadrilateral If The Lengths Of Four Sides And A Diagonal Is Given

Ex. Construct quadrilateral PQRS such that l(PQ) = 5.6 cm, l(QR) = 5 cm, l(PS) = 4.3 cm, l(RS) = 7 cm, l(QS) = 6.2 cm

Solution: Let us draw a rough figure and show the given information in it. From the figure we see that the sides of triangle SPQ and triangle SRQ are known. So if we construct triangle SPQ and triangle SRQ of given measures, we get quadrilateral PQRS. Construct the given quadrilateral on your own.

Teacher's Note

A quadrilateral needs five measurements. Think of making a door frame using wood pieces. You need the length, width, and the diagonal measurement to make it strong.

Exam Trick

Remember: To make a quadrilateral, split it into two triangles using a diagonal. If you know the sides of both triangles, you can draw the quadrilateral easily.

Points To Remember

A quadrilateral has 10 elements: 4 sides, 4 angles, and 2 diagonals.
You need exactly 5 pieces of information to draw a quadrilateral.
Diagonal divides a quadrilateral into two triangles.
You can draw a quadrilateral if you can draw its two triangles.

(II) To Construct A Quadrilateral If Three Sides And Two Diagonals Are Given

Ex. Construct quadrilateral WXYZ such that l(YZ) = 4 cm, l(ZX) = 6 cm, l(WX) = 4.5 cm, l(ZW) = 5 cm, l(YW) = 6.5 cm

Solution: Let us draw a rough figure and show the given measures in it. From the figure we see that all sides of triangle WXZ and triangle WZY are known. So let us draw triangle WXZ and triangle WZY using given measures. We will get quadrilateral WXYZ after drawing segment XY. Construct this quadrilateral on your own.

Teacher's Note

When you know three sides and two diagonals, you have enough information. This is like building a picture frame where you know the lengths of the sides and the diagonal support.

Exam Trick

Remember: If you have three sides and two diagonals, draw one triangle first, then the second triangle. Connect the remaining sides.

Points To Remember

Two diagonals divide a quadrilateral into four triangles.
If you know three sides and both diagonals, draw triangles using these measurements.
The diagonals help you fix the position of all four points.

(III) To Construct A Quadrilateral If Two Adjacent Sides And Any Three Angles Are Given

Ex. Construct quadrilateral LEFT such that l(EL) = 4.5 cm, l(EF) = 5.5 cm, m angle L = 60°, m angle E = 100°, m angle F = 120°

Solution: Let us show the given information in a rough figure. From the figure we see that segment LE of length 4.5 cm can be drawn and after drawing segment EF making an angle of 100° at the point E of segment LE, we get three points L, E and F. Let us draw rays making an angle of 60° at the point L and a ray making an angle of 120° at the point F. The intersection of these two rays is point T. Now you can construct this quadrilateral LEFT.

Teacher's Note

When you know two sides and three angles, you can find the fourth angle because all angles add up to 360°. This helps you complete the quadrilateral using rays and angles.

Exam Trick

Remember: Draw the two given sides first. Then use the angles to draw rays from the end points. Where the rays meet is your fourth point.

Points To Remember

Two adjacent sides mean two sides that share a vertex.
The sum of all angles in a quadrilateral is 360°.
You can find the fourth angle if you know three angles.
Use protractor to draw angles accurately.

(IV) To Construct A Quadrilateral If Three Sides And Two Angles Included By Them Are Given

Ex. Construct quadrilateral PQRS such that l(QR) = 5 cm, l(RS) = 6.2 cm, l(SP) = 4 cm, m angle R = 62°, m angle S = 75°

Solution: Let us draw a rough figure, show the given information in that figure. From the figure we see that after drawing segment QR, if segment RS is drawn making an angle of 62° at the point R, we can get points Q, R and S of the quadrilateral. We will get point P on ray SP at a distance of 4 cm from S, which makes an angle of 75° at point S. We get quadrilateral PQRS of given measure after joining points P and Q. Now you can do this construction.

Teacher's Note

Included angle means the angle between two known sides. It is like opening a book - the angle between the two covers is the included angle.

Exam Trick

Remember: Start with one side. Draw the next side at the correct angle. Then use the third side and its angle to find the fourth point.

Points To Remember

Included angles are angles between two known sides.
Draw sides and angles in order using a protractor and ruler.
The last two points must be joined to complete the quadrilateral.

Practice Set 8.1

1. Construct the following quadrilaterals of given measures.

(1) In quadrilateral MORE, l(MO) = 5.8 cm, l(OR) = 4.4 cm, m angle M = 58°, m angle O = 105°, m angle R = 90°.

(2) Construct quadrilateral DEFG such that l(DE) = 4.5 cm, l(EF) = 6.5 cm, l(DG) = 5.5 cm, l(DF) = 7.2 cm, l(EG) = 7.8 cm.

(3) In quadrilateral ABCD, l(AB) = 6.4 cm, l(BC) = 4.8 cm, m angle A = 70°, m angle B = 50°, m angle C = 140°.

(4) Construct quadrilateral LMNO such that l(LM) = l(LO) = 6 cm, l(ON) = l(NM) = 4.5 cm, l(OM) = 7.5 cm.

Let's Recall

By putting some conditions on sides and angles of a quadrilateral, we get different types of quadrilaterals. You already know two types of quadrilaterals, namely rectangle and square. Now we will study some more properties of these types and of some more types of quadrilaterals through activities.

Rectangle

If all angles of a quadrilateral are right angles, it is called a rectangle.

Among the five elements given to construct a quadrilateral, at least two have to be lengths of adjacent sides. You can construct a quadrilateral if two adjacent sides and three angles are given.

From the definition, we know that all angles of a rectangle are right angles. So if you know two adjacent sides, then you can construct a rectangle.

Teacher's Note

A rectangle is like the shape of your classroom wall or a piece of paper. All four corners make 90-degree angles.

Exam Trick

Remember: All angles in a rectangle are 90°. So you only need two side lengths to draw a complete rectangle.

Points To Remember

All four angles of a rectangle are 90 degrees.
Opposite sides are equal in length.
Both diagonals are equal in length.
Diagonals bisect each other.

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