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.Read More

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