Question

how to draw a triange when two sides and a median is given

Answer 1

The mid-point theorem states the line segment joining the mid-points of any two sides of triangle is parallel to the third side and equal to half its length.
According to the diagram,

DE || AB
DE = (½) AB
DE = (1/2)x 8
DE = 4 cm
According to the diagram,
BD = DC = 3 cm ( D is a mid point of BC)
Steps of construction:
Step 1: Draw line segment BC = 6 cm.
Step 2: Draw the perpendicular bisector of BC.
Step 3: With B as the centre and radius equal to 4.5 cm, draw an arc above line BC.
Step 4: Now with D as the centre radius 4 cm, draw another arc to intersect the previous arc at E. (Since DE = (1/2)AB )
Step 5: Join points C and E. Extend CE.
Step 6: With B as the centre and radius, draw an arc cutting CE produced at A.
Step 7: Join points A, B, points B, E and points E, D.
Alternate method
In quad. ABCD,
BE = ED
AE = EC ( E is the mid point of side AC)
ABCD is a || gm ( Diagonals AC and BD bisect each other)
AB = CD = 8 cm ( Opposite sides of a parallelogram are equal)
In a triangle BCD BC = 6cm , CD = 8 cm and BD = 9 cm.
Alternate method

Steps of construction:
Step 1: Draw line segment BC = 6 cm.
Step 2: With B as the centre and radius equal to 9 cm, draw an arc.
Step 3: With C as the centre and radius equal to 8 cm, draw another arc cutting the previous arc at point D.
Step 4: Join points B, D and points C, D.
Step 5: With D as the centre and radius equal to 4.5cm, draw an arc cutting side BD at E.
Step 6: Join points C and E. Extend CE.
Step 7: With B as the centre and radius 8 cm, draw an arc cutting CE produced at A.
Step 8: Join points A, B.
Construct an equilateral triangle ABC whose altitude is 6 cm .
In an equilateral triangle:
•The three sides are congruent.
AB Cong BC Cong AC
•The three angles are congruent.
Measures of Ang A,Ang B and Ang C = 60
•Each altitude bisects the side to which it is perpendicular, and so it is also the median to that side.
There fore, AD is the median of side BC.
•Each altitude bisects the angle at the vertex it passes through AD bisects ang A .
Ang BAD = ,Ang DAC = 30
In Tri ABD , AD + 6 cm ANG B = 60 and Ang A = 30

Steps of construction:
Step 1: Draw line PQ.
Step 2: Draw a perpendicular to PQ from any point on it, say D. Let this be DX.
Step 3: With D as the centre and radius equal to 6cm, draw an arc cutting ray DX at A.
Step 4: Draw ang DAY = 30 , with ray AY cutting PQ at point B.
Step 5: With B as the centre and radius equal to AB, draw an arc cutting line PQ at point C.
Step 6: Join points A and C.
ABC is the required triangle.
Construct a triangle ABC in which AB = 5 cm , BC = 6 cm and length of Median CD = 4 cm.

Steps of construction:
Step 1: Draw line segment AB = 5 cm.
Step 2: Draw the perpendicular bisector of AB.
Step 3:With D as the centre and radius 4 cm, draw an arc above line AB.
Step 4: With B as the centre and radius 6 cm, draw an arc intersecting the previous arc at C.
Step 5: Join points A, C, points C, D and points B, C.
ABC is the required triangle.
Construct a triangle ABC in which AB = 4 cm, AC = 5 cm and the length of the perpendicular from A on BC is 3 cm.
Step 1: Draw line PQ.
Step 2: Draw ang QDX = 90 .
Step 3: With D as the centre and radius equal to 3cm, draw an arc cutting ray DX at A.
Step 4: With A as the centre and radius equal to 4cm, draw an arc cutting line PQ at point B.
Step 5: With A as the centre and radius equal to 5 cm , mark another arc cutting line PQ at point C.
Step 6: Join points A, B and points A, C.

ABC is the required triangle.