Download Practical Geometry

PRACTICAL GEOMETRY
Introduction:
Practical
Geometry will enable the Student to identify and
classify geometric figures and apply geometric
properties
to
reasoning
and
practical
situations,
visualization
skill
special
will
be
enhanced by using Geometric Models and a
variety of technologies.
The Bisectors of Angles of a given triangle.
Given:
ABC is a triangle
Required : To draw the bisectors of angles A, B and C
Construction:
1.
2.
Draw the ABC
With point B as a centre draw an arc of any radius, intersecting the sides BC and BA at point L and M
3.
Take point L as a centre and draw an arc of any radius.
Now take M as a centre and with the same radius draw the another arc, which cuts the previous arc
at point P
Join point P to B and produce it. Thus BP is the bisector of <B
Similarly, repeat steps (2) to (5) to draw CQ and AR the bisectors of <C and <A respectively.
4.
5.
6.
ANGLES BISECTORS OF  ABC
Hence BP, CQ and AR are the required bisectors of angles of a  arc concurrent.
Note: The angles bisectors intersect at a point I, which is called in centre and the bisectors of angles of a 
are concurrent.
•
How to draw the right bisectors of sides of a triangle ABC.
Given:
A  ABC
Required:
To draw right bisectors of the sides AB, BC
and CA
CONSTRUCTION:
1. Draw the  ABC.
2. To draw right bisector of the side AB, with B as a centre and radius more than half of
AB, draw arcs on either sides of AB.
3. Now with A as a centre and with the same radius, draw arcs on either sides of AB
cutting previous arcs at P and Q.
4. Join P and Q
Thus PQ is the right bisector of AB,
5. Similarly, repeat steps (2) to (4) and draw ST and LM, the right bisectors of BC and AC
respectively. Hence PQ, ST and LM arc the required right bisector of the sides AB, BC
and AC respectively of  ABC.
Note: Right bisectors of ABC pass through O. So they arc concurrent.
RIGHT BISECTORS OF  ABC
COVERAGE OF THE LESSON
ACCORDING TO ITS SYNOPSIS:
1. Construction of Circum Circle
Construct the PQR in which mPQ = mQR = 8cm, m<Q =60°. Draw the “Circum Circle” of
the  PQR.
Circle:
A set of points of a plane which are equidistant from a given (fixed) point is called a CIRCLE:
The fixed point is called the “Centre” of the Circle.
Circum Circle:
A circle which passes through the three vertices of a triangle is called “Circum Circle” (or
circumscribed circle) and the centre is called “Circum Centre” of the circle.
Given:
A  PQR, in which mPQ =mQR = 8cm and m<Q=60°
Required:
Draw the Circum circle of  PQR.
CONSTRUCTION:
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Draw a line segment QR = 8cm.
At point Q draw an <RQX = 60°
From QX, cut off PQ = 8 cm.
Join P to R. Thus  PQR is constructed.
Draw the right bisector LL’ of side QR.
Draw the right bisector MM’ of side PR. Let them intersect each
other at point “O”.
• With Centre “O” and radius equal to OR, draw a circle, which
passes through the vertices of  PQR.
Hence it is our required circum circle is constructed (constructed)
CIRCUM CIRCLE OF  PQR
2. CONSTRUCTION OF
ESCRIBED CIRCLE :
2. Construction of Escribed Circle
Question:
Construct  ABC in which mAB = 7.5cm mBC = 7.5cm, m<B=30°
Draw the escribed circle opposite to Vertex B.
Given:
ABC, in which mAB = mBC = 7.5cm, and m<B=30°
Required:
Draw the escribed circle opposite to Vertex B
CONSTRUCTION:
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Draw a line segment BC = 7.5cm.
At point B, draw an < CBX=30°
From BX, cut off AB = 7.5 cm.
Join A to C. Thus  ABC is constructed.
Produce (extend) BC beyond C to BY
Produce BA to BX. Making two exterior angles <ACY and <CAX.
Draw the bisector AA’ and CC’ of <CAX and <ACY respectively let them intersect each other at point “O”.
From point “O” draw OD perpendicular on BY.
With centre O and radius equal to OD draw a circle. Which touching BA, BY and AC.
This is called the escribed circle opposite to vertex B
ESCRIBED CIRCLE OPPOSITE TO VERTEX “B” OF  ABC
ESCRIBED CIRCLE:
A circle which touches one side of a triangle externally and the other two sides produced internally.
3.
CONSTRUCTION OF IN
CIRCLE
Question:
Construct the  ABC, in which m<B = 60°, mBC = 7.5cm, mAB = 7cm. Draw
the in circle (inscribed circle) of ABC.
Solution
Given:
A  ABC, in which m<B = 60°, mBC = 7.5cm and mAB = 7cm.
Required:
Draw the in circle of ABC.
In-Circle:
The circle which touches the three sides of a triangle is called in-circle or
inscribed circle of the triangle and its centre is called “In-Centre”.
CONSTRUCTION:
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Draw a line segment BC = 7.5cm
At point “B” draw an <CBX = 60°.
From BX, cut of AB = 7cm.
Join A to C. Thus  ABC is constructed.
Draw the bisector BB’ of >B.
Draw the bisector CC’ of >C (et them intersect each other at point O.
From point “O” draw OD perpendicular on BC.
With Centre “O” and radius equal to OD draw a circle. The circle touches
the three sides of ABC
Hence it is our required in-circle is constructed.
4. CONSTRUCTION OF MEDIANS OF A
TRIANGLE
Question:
Construct a  ABC, such that mAB = 6.8cm, mBC = 8cm and
mAC = 10.0cm. Draw the medians of ABC.
Median of a Triangle:
Line segments joining, the vertices of a triangle to the mid
points of the opposite sides are called medians of the triangle.
Given:
A triangle ABC, in which mAB = 6.8cm, mBC = 8cm and mAC
= 10.0cm.
Required:
Draw the medians of ABC.
CONSTRUCTION:
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Draw a line segment BC = 8cm
From point B draw an arc of radius 6.8cm.
From point C draw an arc of radius 10cm which intersects the previous arc at
point A.
Join A to B and A to C.
Thus  ABC is constructed.
Draw the mid point D of Side BC
Draw the mid point E of Side AC
Draw the mid point F of Side AB
Join A to D, B to E and C to F.
Thus AD, BE and CF are our required medians of ABC, which meet in a point
O.
It may be noted that medians of every triangle are concurrent (i.e. meet in one
point) and their point of concurrency called “centraid”, divides each of them in
the ratio of 2:1.
By actual measurement it can be prove that
mAO = mBO = mCO = 2
mOD mOE
mOF
1
MEDIANS OF  ABC
5. CONSTRUCTION OF
ALTITUDES OF A TRIANGLE
Question:
Take any triangle ABC and draw its altitudes.
Given:
A  ABC
Required:
To draw altitudes of the  ABC
Altitude:
A perpendicular from the vertex of a triangle to the
opposite side is called an altitude of the triangle.
CONSTRUCTION:
1. Draw a  ABC
2. Take a point A as centre and draw an arc of suitable radius,
which cuts BC at points D and E.
3. From D as centre, draw an arc of radius more than ½ mDE.
4. Again from point E draw another arc of same radius, cutting
first arc at point F.
5. Join the points A and F, such that AF intersects BC at point P.
Then AP is the altitude of the  ABC from the vertex A.
6. Similarly, repeat the steps (2) to (5) and draw BQ and CR, the
altitudes of  ABC from the vertices B and C respectively.
Hence AP, BQ and CR are the required altitudes of  ABC, which
meet in a point O.
The points P, Q and R are called the feet of these altitudes.
Note: It may be remember that the altitudes of a triangle are
always concurrent. The point of concurrency is called the
“ORTHO CENTRE”
ALTITUDES OF  ABC
SUMMARY OF THE LESSON:
1)
A circle which passes through the vertices of a triangle is called circumcircle.
2)
A circle which touches the three sides of a triangle is called In-Circle.
3)
The line segments joining the vertices of a triangle with the mid-points of
the opposite sides are called “Medians” of the triangle.
4)
The point of intersection of three medians is called “Centroid” and medians
are concurrent.
5)
The line segments which are perpendiculars from the vertices to the
opposite sides of a triangle are called its “Altitudes”.
6)
The point of intersection of three altitudes is called “Ortho Centre” and
they are concurrent.
7)
Construction of unique triangle is possible when the sum of measures of
two sides is greater than the third side.
ACTIVITIES (M.C.Qs)
1)
The sum of measures of all the angles of a triangle is __________ degree.
(a) 180 (b) 360 (c) 90
2)
Two lines are said to be perpendicular on each other if they form a/an _____ angle.
(a) acute (b) Obtuse
3)
(d) 270
(c) Reflex (d) Right
The right bisectors of a triangle are ________
(a) Concurrent (b) Not concurrent
(c) Concurrent and passes through the same point
4)
(d) None
The bisectors of angles of a triangle are and the point of concurrency is called
__________
(a) Incentre (b) Ortho Centre
(c) Circum Centre (d) Centroid.
5)
A circle which passes through the vertices of a triangle is called __________ .
(a) In-circle (b) Circum-Cricle
(c) Escribed Circle
Page -2-
6)
A circle which touches the three sides of a triangle is called __________
(a) Circum circle (b) Escribed circle
(c) In-Circle
7)
The point of intersection of three medians is called _____ .
(a) Ortho Centre (b) Centroid
(c) None (d) In-Centre
8)
The point of intersection of three altitudes is called ________
(a) Centroid (b) Ortho Centre
(c) In-Centre (d) Curcum Centre
9)
The line segments joining the vertices of a triangle with the mid-points of the
opposite sides are called __________
(a) Altitudes
(b) Medians
(c) Perpendiculars (d) None
10)
The line segments which are perpendiculars from the vertices to the opposite
sides of a triangle are called ________
(a) Medians
(b) Altitudes (c) Bisectors (d) None
MCQs Answer Key:
1-
(a) 180
2.
same point.
4-
(a) In-Centre 5. (b) Circum Circle
6.
(c) In-Circle
8.
(b) Ortho centre
10. (b) Altitudes
(d) Right
3. (c) Concurrent and passes through the
7. (b) Centroid
9. (b) Medians
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