Tiêu đề 03-Practical Geometry(1).pdf ✅

PRACTICAL GEOMETRY 3 CHAPTER CONTENTS • Construction of a pair of parallel lines using set squares and ruler • Construction of parallel lined by ruler & compasses • Triangle construction (Possibilities) • Side-Side -Side (SSS) Triangle construction • Side-Angle-Side (SAS) Triangle construction • Angle-Side-Angle (ASA) Triangle construction • Construction of a right angled triangle when its hypotenuse and one side are given ➢ CONSTRUCTION OF A PAIR OF PARALLEL LINES USING SET SQUARES AND RULER Construct a line parallel to a given line l passing through a given point P not on the line : (i) Draw a line l. (ii) Place a set square with the arm of its right angle. (iii) Holding the set square fixed, place a ruler along the other arm of the right angle. (iv) Holding the ruler fixed, slide the set square along the edge of the ruler until the perpendicular edge of the set square passes through the given point P as shown in figure 0cm 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 2 1 1 2 3 4 5 6 7 P l 0cm 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 2 1 1 2 3 4 5 6 7 P Q l P Q l (v) Keeping the set square fixed at same position, draw a line PQ through the point P along the edge of set square. The line PQ is the required line through the point P and parallel to the line l.

Then, m is the required line parallel to l and passing through the given point A. Note : In the above figure, l and m are two parallel lines and AB is a transversal. ABC and FAB are alternate interior angles But ABC = FAB [By construction] Hence, l || m [ alternate interior angles are equal]  Alternative Method S.No. Steps of Construction Construction 1. Draw a line l. l 2. Mark a point A which is not lying on l. A l 3. Draw a line n, through A which meets the line l at point B A B l C n 4. At point A, draw an angle such that it is equal to ABC. A B l C 5. Draw line m which is parallel to line l. A B l m C ➢ TRIANGLE CONSTRUCTION (POSSIBILITIES) There are following possibilities of constructing a triangle : (1) When its three sides are given. (Also known as SSS triangle construction). (2) When its two sides and the included angle is given. (Also known as SAS triangle construction). (3) When its two angles and the included side is given. (Also known as ASA triangle construction). • The sum of two sides of a triangle is always greater than the third side. • The sum of three angles of a triangle should be equal to 180°. ➢ SIDE-SIDE-SIDE (SSS) TRIANGLE C CONSTRUCTION When length of the three sides of a triangle are given, we follow the following steps for constructing a triangle : S.No. Steps of Construction Construction 1. Draw a line segment (with the help of ruler), of a given length equal to one side of a triangle. Let us name it as AB. A B 5 cm 2. From one end point A, draw an arc whose distance from A is equal to second side. A B 5 cm 3. From second end point B, draw another arc whose distance from B is equal to third side and which cuts the first arc at a point C. A B 5 cm C 4. Join AC and BC. C 7 cm 6 cm 5 cm A B The triangle so obtained is the required triangle. It is important to note that the sum of two sides of a triangle is always greater than the third side.
Thus we can say that we cannot construct a triangle when the sum of any two sides is less than or equal to the third side. ❖ EXAMPLES ❖ Ex.1 Construct a triangle ABC such that side AB = 5 cm, BC = 6 cm and AC = 7 cm. Sol. C 7 cm 6 cm 5 cm A B 1. Draw a line segment AB of length 5 cm. 2. With centre A and radius 7 cm draw an arc of the circle. 3. With centre B and radius 6 cm draw another arc intersecting first arc at C. 4. Join AC and BC as shown in figure. ABC is the required triangle. Ex.2 Construct a triangle ABC in which AB = 4.5 cm, AC = 6 cm, BC = 5 cm. Sol. C 6 cm 5 cm 4.5 cm A B 1. Draw a line segment AB of length 4.5 cm. 2. With centre A and radius 6 cm draw an arc of the circle. 3. With centre B and radius 5 cm draw another arc intersecting the first arc at C. 4. Joint AC and BC as shown in figure. ABC is the required triangle. Ex.3 Construct a triangle ABC where AB = 2 cm, BC = 3 cm and AC = 6 cm. Sol. Before constructing the required triangle, let us first draw a rough sketch of the triangle. C 6 cm 3 cm 2 cm A B By plotting the rough sketch, we find that it is not possible to construct this triangle. Because the sum of any two sides of a triangle is always greater than third side but in the given triangle, the sum of the lengths of the two sides AB (2 cm) and BC (3 cm) is less than the length of the third side AC (6 cm). AB + BC < AC Therefore, the three given sides of the triangle do not satisfy the triangle inequality. Hence, it is not possible to construct the given triangle. ➢ SIDE-ANGLE-SIDE (SAS) TRIANGLE C CONSTRUCTION When the length of two sides and the measure of included angle is given, we follow the following steps for constructing a triangle. S.No. Steps of Construction Construction 1. Draw a line segment AB of the triangle with help of ruler. A B 5 cm 2. Draw OBA of measure equal to the given B. 5 cm A B O 30° 3. From any point on BO , cut off line segment equal to second side. Mark the cut off point as C. 5 cm A B O 30° C 3 cm 4. Join AC. The triangle so obtained is the required triangle. 5 cm A B O 30° C 3 cm