Download Practical Geometry

Practical Geometry
Main Concepts and Results:
• Five measurements can determine a quadrilateral uniquely.
• A quadrilateral can be constructed uniquely if the lengths of its four sides and a
diagonal are given.
• A quadrilateral can be constructed uniquely if the lengths of its three sides and two
diagonals are given.
• A quadrilateral can be constructed uniquely if its two adjacent sides and three angles
are given.
• A quadrilateral can be constructed uniquely if its three sides and two included angles are
given.
Exercise:
1) We require three measurements (of sides and angles) to draw a unique triangle. True or
false?
2) Arshad has five measurements of a quadrilateral ABCD. These are AB = 5 cm, ∠A = 50°, AC =
4 cm, BD = 5 cm and AD = 6 cm. Can he construct a unique quadrilateral? Give reasons for
your answer.
3) How to construct a unique quadrilateral given the following measurements:
(I)
When four sides and one diagonal are given.
(II)
When two diagonals and three sides are given.
(III)
When two adjacent sides and three angles are given.
(IV)
When three sides and two included angles are given.
(V)
When other special properties are known.
Explain the methods.
4) Construct a quadrilateral PQRS where PQ = 4 cm, QR = 6 cm, RS = 5 cm, PS = 5.5 cm and PR =
7 cm.
5) We know that 5 measurements of a quadrilateral can determine a quadrilateral uniquely. Do
you think any five measurements of the quadrilateral can do this?
6) Can you draw a parallelogram BATS where BA = 5 cm, AT = 6 cm and AS = 6.5 cm? Why?
7) Can you draw a rhombus ZEAL where ZE = 3.5 cm, diagonal EL = 5 cm? Why?
8) A student attempted to draw a quadrilateral PLAY where PL = 3 cm, LA = 4 cm, AY = 4.5 cm,
PY = 2 cm and LY = 6 cm, but could not draw it. What is the reason?
9) Construct the following quadrilaterals.
10) Construct a quadrilateral ABCD, given that BC = 4.5 cm, AD = 5.5 cm, CD = 5 cm the diagonal
AC = 5.5 cm and diagonal BD = 7 cm.
11) In the above question, can we draw the quadrilateral by drawing ∆ ABD first and then find
the fourth point C?
12) Can you construct a quadrilateral PQRS with PQ = 3 cm, RS = 3 cm, PS = 7.5 cm, PR = 8 cm
and SQ = 4 cm? Justify your answer.
13) Construct the following quadrilaterals.
14) Construct a quadrilateral MIST where MI = 3.5 cm, IS = 6.5 cm, ∠M = 75°, ∠I = 105° and ∠S =
120°.
15) Can you construct the above quadrilateral MIST if we have 100° at M instead of 75°?
16) Can you construct the quadrilateral PLAN if PL = 6 cm, LA = 9.5 cm, ∠P = 75°, ∠L =150° and
∠A = 140°? (Hint: Recall angle-sum property).
17) In a parallelogram, the lengths of adjacent sides are known. Do we still need measures of the
angles to construct as in the example above?
18) Construct the following quadrilaterals.
19) Construct a quadrilateral ABCD, where AB = 4 cm, BC = 5 cm, CD = 6.5 cm and ∠B = 105° and
∠C = 80°.
20) In the above question, we first drew BC. Instead, what could have been be the other starting
points?
21) We used some five measurements to draw quadrilaterals so far. Can there be different sets
of five measurements (other than seen so far) to draw a quadrilateral? The following
problems may help you in answering the question.
(i)
Quadrilateral ABCD with AB = 5 cm, BC = 5.5 cm, CD = 4 cm, AD = 6 cm and ∠B =
80°.
(ii)
Quadrilateral PQRS with PQ = 4.5 cm, ∠P = 70°, ∠Q = 100°, ∠R = 80° and ∠S = 110°.
(iii)
Construct a few more examples of your own to find sufficiency/insufficiency of the
data for construction of a quadrilateral.
22) Construct the following quadrilaterals.
23) Draw a square of side 4.5 cm.
24) Is it possible to construct a rhombus ABCD where AC = 6 cm and BD = 7 cm? Justify your
answer.
25) How will you construct a rectangle PQRS if you know only the lengths PQ and QR?
26) Construct the kite EASY if AY = 8 cm, EY = 4 cm and SY = 6 cm (Fig 4.26). Which properties of
the kite did you use in the process?
27) Draw the following.
28) Construct a rhombus PAIR, given that PA = 6 cm and angle ∠A = 110°.
29) Construct a quadrilateral ABCD in which AB = AD = 5cm, BC = CD = 7cm and BD = 6cm. What
type of quadrilateral is this?
30) Construct a trapezium ABCD in which AB||DC, ∠A = 105°, AD = 3 cm, AB = 4 cm and CD = 8
cm.
31) Construct a parallelogram ABCD in which AB = 4 cm, BC = 5 cm and ∠B = 60°.
32) Construct a rhombus whose side is 5 cm and one angle is of 60°.
33) Construct a rectangle whose one side is 3 cm and a diagonal equal to 5 cm.
34) Construct a square of side 4 cm.
35) Construct a rhombus CLUE in which CL = 7.5 cm and LE = 6 cm.
36) Construct a quadrilateral BEAR in which BE = 6 cm, EA = 7 cm, RB = RE = 5 cm and BA = 9 cm.
Measure its fourth side.
37) Construct a parallelogram POUR in which, PO=5.5 cm, OU = 7.2 cm and ∠O = 70°.
38) Draw a circle of radius 3 cm and draw its diameter and label it as AC. Construct its
perpendicular bisector and let it intersect the circle at B and D. What type of quadrilateral is
ABCD? Justify your answer.
39) Construct a parallelogram HOME with HO = 6 cm, HE = 4 cm and OE = 3 cm.
40) Is it possible to construct a quadrilateral ABCD in which AB = 3 cm, BC = 4 cm, CD = 5.4 cm,
DA = 5.9 cm and diagonal AC = 8 cm? If not, why?
41) Is it possible to construct a quadrilateral ROAM in which RO=4 cm, OA = 5 cm, ∠O = 120°, ∠R
= 105° and ∠A = 135°? If not, why?
42) Construct a square in which each diagonal is 5cm long.
43) Construct a quadrilateral NEWS in which NE = 7cm, EW = 6 cm, ∠N = 60°, ∠E = 110° and ∠S =
85°.
44) Construct a parallelogram when one of its side is 4cm and its two diagonals are 5.6 cm and
7cm. Measure the other side.
45) Construct a trapezium RISK in which RI||KS, RI = 7 cm, IS = 5 cm, RK=6.5 cm and ∠I = 60°.
46) Construct a trapezium ABCD where AB||CD, AD = BC = 3.2cm, AB = 6.4 cm and CD = 9.6 cm.
Measure ∠B and ∠A.
[Hint : Difference of two parallel sides gives an equilateral triangle.]
47) State whether the statements are true (T) or (F) false.
(I)
A quadrilateral can be drawn if only measures of four sides are given.
(II)
A quadrilateral can be drawn if all four sides and one diagonal is known
(III)
A quadrilateral can be drawn when all the four angles and one side is given.
(IV)
A quadrilateral can be drawn if all four sides and one angle is known.
(V)
A quadrilateral can be drawn if three sides and two diagonals are given.
(VI)
If diagonals of a quadrilateral bisect each other, it must be a parallelogram.
(VII)
A quadrilateral can be constructed uniquely if three angles and any two sides are
given.
(VIII) A parallelogram can be constructed uniquely if both diagonals and the angle
between them is given.
(IX)
A rhombus can be constructed uniquely if both diagonals are given.