Angle `ABC=60^(@) and BA=BC=8` cm. The mid-points of BA and BC are M and N respectively. Draw and describe the locus of a point which is :
4 cm from N.
Mark the point P, which is 4 cm from both M and N, and equidistant from BA and BC. Join MP and NP, and describe the figure BMPN.

Answer

To solve the problem step by step, we will follow the instructions given in the question and the video transcript.

### Step 1: Draw Triangle ABC
1. Draw a line segment BC = 8 cm.
2. At point B, construct an angle of 60 degrees.
3. From point B, measure 8 cm along the angle to locate point A.
4. Connect points A and C to complete triangle ABC.

### Step 2: Find Midpoints M and N
1. To find the midpoint M of line segment AB, use the compass to draw arcs above and below the line from points A and B.
2. Mark the intersection points of the arcs and draw a line through them to find M.
3. Repeat the same process for line segment BC to find midpoint N.

### Step 3: Draw the Locus of Points 4 cm from N
1. With N as the center, draw a circle with a radius of 4 cm. This circle represents all points that are 4 cm away from N.

### Step 4: Mark Point P
1. Point P is 4 cm from both M and N. Use the compass to draw an arc of radius 4 cm from M that intersects the circle drawn around N.
2. Mark the intersection point as P.

### Step 5: Ensure P is Equidistant from BA and BC
1. To ensure that point P is equidistant from lines BA and BC, draw the angle bisector of angle ABC.
2. Point P should lie on this bisector, confirming it is equidistant from both lines.

### Step 6: Join MP and NP
1. Draw line segments MP and NP to connect point P to midpoints M and N.

### Step 7: Describe Figure BMPN
1. In quadrilateral BMPN, all sides BM, MP, PN, and NB are equal to 4 cm.
2. Since MP is parallel to BC and NP is parallel to AB (using the midpoint theorem), we can conclude that BMPN is a rhombus.

### Final Description
- The figure BMPN is a rhombus because:
- All sides are equal (BM = MP = PN = NB = 4 cm).
- Opposite sides are parallel (MP || BC and NP || AB).