04 - Let's Recap - Constructions - Class 10 - Maths Class 10 Notes | EduRev

Crash Course for Class 10 Maths by Let's tute

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Class 10 : 04 - Let's Recap - Constructions - Class 10 - Maths Class 10 Notes | EduRev

 Page 1


  
  
CONSTRUCTIONS 
 
 
Construction is very important in geometry which helps to develop the skill of 
drawing figures accurately.  
 
Division of a line segment:  
 
In order to divide a line segment internally in a given ratio m: n, where both m and 
n are positive integers, the following steps are followed: 
 
Given: A line segment PQ and ratio m: n. 
Required: To divide line segment PQ in the ratio m: n. 
Steps of construction:  
 Draw a line segment PQ of given length by using a ruler. Draw any ray PX 
making a suitable acute angle with PQ. 
 Along PX draw (m + n) arcs intersecting the rays PX at P
1
, P
2
,.....P
m
....P
m+n.
 
 Join the points i.e. Q P
m+n
. 
 Draw a line through point  P
m 
 parallel to P
m+n
 . 
 We get one point A; this point divides PQ internally in the ratio m: n. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 2


  
  
CONSTRUCTIONS 
 
 
Construction is very important in geometry which helps to develop the skill of 
drawing figures accurately.  
 
Division of a line segment:  
 
In order to divide a line segment internally in a given ratio m: n, where both m and 
n are positive integers, the following steps are followed: 
 
Given: A line segment PQ and ratio m: n. 
Required: To divide line segment PQ in the ratio m: n. 
Steps of construction:  
 Draw a line segment PQ of given length by using a ruler. Draw any ray PX 
making a suitable acute angle with PQ. 
 Along PX draw (m + n) arcs intersecting the rays PX at P
1
, P
2
,.....P
m
....P
m+n.
 
 Join the points i.e. Q P
m+n
. 
 Draw a line through point  P
m 
 parallel to P
m+n
 . 
 We get one point A; this point divides PQ internally in the ratio m: n. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
Construction of a triangle similar to a given triangle: 
Suppose we are given ABC ? and we have to construct a triangle whose sides are 
equal to 
n
m
of the corresponding sides of ABC ? . 
 If m > n, then the triangle to be constructed is larger than the given triangle. 
 If m < n, then the triangle to be constructed is smaller than the given 
triangle. 
Scale Factor: Scale factor means the ratio of the sides of the triangle to be 
constructed with corresponding sides of the given triangle. 
 
Construction of tangents to a circle: 
 
1) To draw tangents to a circle from a point outside it, when centre of the  
      Circle is known: 
Given: A circle is given with centre as O and radius r, and an external point A  
Required: To draw the tangents to the circle from the point A. 
Steps of construction: 
 Join OA, and bisect it. 
 Let R be the midpoint of OA. 
 Taking point R as centre and RO as radius draw a circle to intersect the 
given circle at two points X and Y. 
 Draw a rays AX and AY. 
 Ray AX and AY are the required tangents from A to given circle. 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 3


  
  
CONSTRUCTIONS 
 
 
Construction is very important in geometry which helps to develop the skill of 
drawing figures accurately.  
 
Division of a line segment:  
 
In order to divide a line segment internally in a given ratio m: n, where both m and 
n are positive integers, the following steps are followed: 
 
Given: A line segment PQ and ratio m: n. 
Required: To divide line segment PQ in the ratio m: n. 
Steps of construction:  
 Draw a line segment PQ of given length by using a ruler. Draw any ray PX 
making a suitable acute angle with PQ. 
 Along PX draw (m + n) arcs intersecting the rays PX at P
1
, P
2
,.....P
m
....P
m+n.
 
 Join the points i.e. Q P
m+n
. 
 Draw a line through point  P
m 
 parallel to P
m+n
 . 
 We get one point A; this point divides PQ internally in the ratio m: n. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
Construction of a triangle similar to a given triangle: 
Suppose we are given ABC ? and we have to construct a triangle whose sides are 
equal to 
n
m
of the corresponding sides of ABC ? . 
 If m > n, then the triangle to be constructed is larger than the given triangle. 
 If m < n, then the triangle to be constructed is smaller than the given 
triangle. 
Scale Factor: Scale factor means the ratio of the sides of the triangle to be 
constructed with corresponding sides of the given triangle. 
 
Construction of tangents to a circle: 
 
1) To draw tangents to a circle from a point outside it, when centre of the  
      Circle is known: 
Given: A circle is given with centre as O and radius r, and an external point A  
Required: To draw the tangents to the circle from the point A. 
Steps of construction: 
 Join OA, and bisect it. 
 Let R be the midpoint of OA. 
 Taking point R as centre and RO as radius draw a circle to intersect the 
given circle at two points X and Y. 
 Draw a rays AX and AY. 
 Ray AX and AY are the required tangents from A to given circle. 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
2) To draw tangents to a circle from a point outside it, when centre of the 
circle is not known: 
 
Given: A circle and a point V outside it 
Required: To draw tangents from point V to the circle. 
Steps of construction: 
 Draw a secant VXY to intersect the circle at two points X and Y. 
 Extend VX to point Z, such that VX= VZ, 
 With ZY as diameter draw semi-circle.  
 Draw VW perpendicular to ZY, intersecting semicircle drawn at W. 
 Taking VW as radius and V as centre draw an arcs intersecting given circle 
at R and R’ 
 Draw rays VR and VR’ 
 Rays VR and VR’ are the required tangents. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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