Practical Geometry Class 8 Notes- Chapter 4

Construction of a Quadrilateral

It is very easy to construct a quadrilateral when its five measurements are determined that is

  • The length of the four sides and the length of its diagonal is known
  • The length of the three sides and the length of the two diagonals are known
  • If the three angles and two adjacent sides are given
  • If the three sides and two angles are given

4 Sides and 1 Diagonal

Construction of a Quadrilateral when different measures of sides and angles are given

A unique quadrilateral can be constructed when the following measurements are given:

  • Four sides and one diagonal.
  • Two diagonals and three sides.
  • Two adjacent sides and three angles.
  • Three sides and two included angles.
  • When other special properties are known.

SSS Construction

To construct a ABC, the length of whose sides are, AB = x cm, BC = y cm, and AC = z cm, we will do it in the following manner:

  • Construct a line segment AB, whose length is x cm.
  • With A as the center, draw an arc of radius z cm.
  • With B as the center, draw an arc of radius y cm on the same side. The point where the arcs intersect is the required point C.
  • Join AC and BC.

ABC is the required triangle.

Construction of a Quadrilateral when four sides and one diagonal are given

Suppose we have to construct a quadrilateral PQRS, where PQ = 4 cm, QR = 6 cm, RS = 5 cm, PS = 5.5 cm and PR = 7 cm.
Step 1: Draw a rough sketch to visualize the quadrilateral.

Step 2: Draw PQR as it can be constructed using SSS construction condition.

Step 3: Now we have to locate S, which is at a distance of 5.5 cm from P and 5 cm from R. Also it will be on the opposite side of Q.
With P as center draw an arc of radius 5.5 cm. With R as center draw an arc of radius 5 cm.
S is the point of intersection of the two arcs.

Step 4: Join PS and RS. PQRS is the required quadrilateral.

3 Sides and 2 Diagonals

Construction of a Quadrilateral when two diagonals and three sides are given

Construct a quadrilateral ABCD given, AB = 7 cm, AD = 6 cm, AC = 7 cm, BD = 7.5 cm and BC = 4 cm.
[Make a rough figure for your reference]
Steps of construction of the quadrilateral:

Step 1: ABC can be drawn by SSS construction condition since all its sides are known.
Step 2: With A as center and radius 6 cm (AD), draw an arc.
Step 3: With B as center and radius 7.5 cm (BD) draw another arc to cut the previous arc at D
Step 4: Join AD, BD, and CD.
ABCD is the required quadrilateral

2 Adjacent Sides and 3 Angles

Construction of a Quadrilateral when two adjacent sides and three angles are given

Construct a quadrilateral ALPN, where AL = 6.5 cm, LP = 4 cm, NAL = 110ALP = 75 and LPN = 90.
[Draw a rough Sketch for your reference]:
Steps of construction of the quadrilateral:
Step 1: Draw the line segment AL of length 6.5 cm.
Step 2: Make ALY = 75 at L.
Step 3: Make LAX = 110 at A.
Step 4: With L as center and radius equal to 4 cm, cut an arc on the ray LY at P.
Step 5: Make LPZ = 90 at P.
Step 6: Name the point of intersection of rays PZ and AX as N.
ALPN is the required quadrilateral.

3 Sides and 2 Included Angles

Construction of a Quadrilateral when three Sides and two included angles are given

Construct a quadrilateral ABCD, Where AB = 4.5 cm; BC = 3.5 cm, CD = 5 cm ABC = 45, BCD = 150
[Make a rough figure for your reference]
Steps of construction of the quadrilateral:
Step 1: Draw a line segment BC of length 3.5 cm.
Step 2: Make LBC =  45.
Step 3: Make BCM = 150.
Step 4: With B as center and radius equal to 4.5 cm, cut an arc on the ray LB at A.
Step 5: With C as the center and radius equal to 5 cm, cut an arc on the ray CM at D.
Step 6: Join AD.
ABCD is the required quadrilateral.

Special Quadrilaterals

Construction of a Quadrilateral When Other Special Properties Are Known

Construct a rhombus PQRS with diagonals PR = 5.2 cm and QS = 6.4 cm
[Make a rough figure for your reference]

Note: Diagonals of a rhombus are perpendicular bisectors of each other.

Steps of construction of the Rhombus:
Step 1: Draw a line segment PR of length 5.2 cm.
Step 2: Draw the perpendicular bisector of PR. Name the point O, where the perpendicular bisector of PR and PR intersect.
Step 3: With O as center and radius equal to 3.2 cm cut arcs on both sides of the perpendicular bisector. Name them as Q and S.
Step 4: Join, PQ, QR, RS, and PS.

PQRS is the required rhombus.

Introduction to Practical Geometry

Number of measurements necessary for construction of a unique Quadrilateral

To draw a unique quadrilateral we need at least five measurements of sides and anglesHowever, it is not necessary that we will get a unique quadrilateral if we have the measurements of any five combinations of sides and angles.

For example, a unique quadrilateral can be drawn if we are given the measurement of four sides and one diagonal of a quadrilateral.
However, a unique quadrilateral will not be drawn if we are given the measurement of two diagonals and three angles of a quadrilateral.

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