Using a ruler, a pencil and a pair of compasses only, construct a rhombus LMNO such that line LM = 6cm, angle LMN = 135°. Measure diagonal LN.

Draw a straight line using a ruler. Mark a point L on it. From L, measure 6 cm along the line and mark point M. So, $\text{LM} = 6 \text{ cm}$.

• Extend the line segment LM to the right of M.
• Place the compass at M and draw a semicircle that intersects LM (extended) at two points. Let's call the point on the extension P.
• From P, with the same compass radius, mark an arc on the semicircle (this is $60^\circ$). From this new point, mark another arc (this is $120^\circ$). From this $120^\circ$ mark, draw another arc.
• Construct a perpendicular line at M (a $90^\circ$ angle). To do this, from the $60^\circ$ and $120^\circ$ marks on the semicircle, draw arcs that intersect above M. Draw a line from M through this intersection. This line forms a $90^\circ$ angle with LM.
• The angle between the $90^\circ$ line and the extended line (to the right of M) is $90^\circ$. Bisect this $90^\circ$ angle. The bisector will form an angle of $45^\circ$ with the $90^\circ$ line.
• The ray for $135^\circ$ will be the line formed by adding this $45^\circ$ to the $90^\circ$ line, measured from LM. So, the angle $\angle \text{LMN} = 90^\circ + 45^\circ = 135^\circ$.

• With M as the center and a radius of 6 cm (since all sides of a rhombus are equal), draw an arc that intersects the $135^\circ$ ray constructed in Step 2. Mark this intersection point as N. So, $\text{MN} = 6 \text{ cm}$.

• With L as the center and a radius of 6 cm, draw an arc above LM.
• With N as the center and a radius of 6 cm, draw another arc that intersects the previous arc. Mark this intersection point as O.

• Join points L to O and N to O with straight lines. You have now constructed the rhombus LMNO.

• Draw a straight line connecting point L to point N.
• Measure the length of the line segment LN using a ruler.

Using the Law of Cosines for triangle LMN:

The measured length of diagonal LN is approximately $\boxed{11.1 \text{ cm}}$.

2 zimekamilika. Zimebaki 3 leo — tuma inayofuata.