Construction of a triangle given a base and base angle and the difference of other two sides


 
 
Concept Explanation
 

Construction of a triangle given a base and base angle and the difference of other two sides

Construction of a triangle given a base and base angle and the difference of other two sides : In order to contruct a triangle when its base, difference of the other two sides and one of the base angles are given, we follow the following steps:

Steps of Contruction: Obtain the base, base angle and the difference of two other sides. Let BC be the base, dpi{100} large angle B be the base angle and x be the difference of the other two sides AB and AC of large Delta ABC .. There can be two cases

Case I: AB > AC  i.e., x =AB - AC

STEP I   Draw the base BC of given length.

STEP II Draw large angle CBX of measure equal to that of large angle B.

STEP III  AS AB > AC , then cut off segment BD = AB - AC from ray BX

STEP IV . Join DC

STEP V: From point C taking radius more than half of CD Draw arc on both side of CD.
STEP VI : From point D taking same radius as in step V Draw arc on both side of CD to intersect the arcs drawn in step V at point P and Q.
STEP VII :  Join PQ . This is the perpendicular bisector of CD meeting BX at A.
STEP VIII Join AC to obtain the required triangle ABC.

Case II: AC > AB  i.e., x =AC - AB

STEP I   Draw the base BC of given length.

STEP II Draw large angle CBX of measure equal to that of large angle B.

STEP III  As AC > AB ,extend XB to D such that BD = AC - AB

STEP IV . Join DC

STEP V: From point C taking radius more than half of CD Draw arc on both side of CD.
STEP VI : From point D taking same radius as in step V Draw arc on both side of CD to intersect the arcs drawn in step V at point P and Q.
STEP VII :  Join PQ . This is the perpendicular bisector of CD meeting BX at A.
STEP VIII Join AC to obtain the required triangle ABC.

 

Justification: Let us now see how do we get the required triangle. Since A lies on the perpendicular bisector of DC.

large therefore       AD = AC

So,    BD = AD - AB = AC - AB

Example : Construct a triangle ABC in which base AB = 5 cm, large angle A=30^{circ} and AC - BC = 2.5 cm.

SOLUTION  In order to construct the triangle ABC, we follow the following steps:

Steps of Construction:

STEP I  Draw base AB = 5 cm

STEP II Draw large angle BAX=30^{circ}

STEP III  From ray AX, cut off line segment AD = 2.5 cm(=AC-BC)

STEP IV  Join BD.

STEP V  Draw the perpendicular bisector of BD which cuts AX at C.

STEP VI  Join BC to obtain the required triangle ABC.

Justification: Since C lies on the perpendicular bisector of DB. THerefore,

              CD = CB

Now,       AD = 2.5 cm

large Rightarrow         AC - CD = 2.5 cm

large Rightarrow        AC - BC = 2.5 cm

Hence, large Delta ABC  is the required triangle.

 
 


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