How To construct a triangle, given its base, a base angle and sum of other two sides

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

Steps of Construction:  Obtain the base, base angle and the sum of other two sides. Let BC be the base,

STEP I Draw the base BC. STEP II Draw of measure given in the question. STEP III  From ray through B making the specified angle, cut-off line segment AX equal to x (the sum of other two sides).
STEP IV Join XC.
STEP V: From point C taking radius more than half of CX Draw arc on both side of CX.
STEP VI : From point X taking same radius as in step V Draw arc on both side of CX to intersect the arcs drawn in step V at point P and Q.
STEP VII :  Join PQ . This is the perpendicular bisector of CX 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 point A lies on the perpendicular bisector of CX.Therefore,      AX = AC

Now,     BA = BX  - AX

      BX = BA + AC

PROOF

STEP I  Obtain the base, base angle and the sum of other two sides. Let AB be the base,

STEP II  Draw the base AB.

STEP III  Draw

STEP IV From ray AX, cut-off line segment AD equal to l ( the sum of other two sides).

STEP V Join BD.

STEP VI Construct an angle

STEP VII Suppose BY intersects AX at C.

    Then, is the required triangle.

Justification: Let us now see how do we get the required triangle.

In

Now,     AC = AD - CD

       AC = AD - BC

      AD = AC + BC

Example : Construct a triangle ABC in which AB = 5.8 cm , BC + CA = 8.4 cm and

SOLUTION  In order to construct the we follow the following steps:

Steps of Construction

STEP I  Draw AB = 5.8 cm

STEP II Draw

STEP III From ray BX, cut off line segment BD = BC + CA = 8.4 cm.

STEP IV Join AD

STEP V  Draw the perpendicular bisector of AD meeting BD at C.

STEP VI  Join AC to obtain the required triangle ABC.

Justification: Clearly, C lies on the perpendicular bisector of AD.

      CA = CD

Now,  BD = 8.4 cm

     BD + CD = 8.4 cm

    BD + CA = 8.4 cm

Hence, is the required triangle.

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1) Construct a Triangle, when its base, sum of the other two sides and one base angle are given.

Step 1: Draw a line segment AB 5.8 cm.

Step 2 : Draw ∠B = 45°.

Step 3 : With Center B and radius 8.4 cm, make an arc which intersects BX at D.

Step 4 : Join D to A.

Step 5 : Draw a perpendicular bisector of segment DA it intersect the line segment BD at point C.

Step 6 : Join C to A.

2) Triangle construction, when its base, difference of the other two sides and one base angle are given.

Step 7 : Join A to C, ABC is the required triangle.

3)Construction of a Triangle of given perimeter,and two base Angles.

AB + BC + CA = 12 cm ∠B = 450 and ∠C= 60°.

Step 1 : Draw a line segment XY of 12 cm

Step 2 : From point X draw ray XD at 45 degree and from point Y draw ray YE at 60 degree

Step 3 : draw angle bisector of X and Y, two angle bisectors intersect each other at point A

Step 4 : Draw line bisector of XA and AY respectively these two line bisectors intersect XY at point B and C

Step 5 : Join A to B and A to C

Step 6 : Triangle ABC is the required triangle.