Draw the base DC.

Draw a straight line and mark a point D on it. Using your compass, open it to $10 \text{ cm}$. Place the compass point at D and draw an arc to intersect the line. Label the intersection point C. So, $DC = 10 \text{ cm}$.

• Place the compass at D and draw a semicircle that intersects DC (extended if necessary) at a point, let's call it X.
• With the same compass opening, place the compass at X and draw an arc to intersect the semicircle at a point, say Y ($60^\circ$).
• From Y, draw another arc to intersect the semicircle at Z ($120^\circ$).
• Bisect the angle between X and Z to construct a $90^\circ$ line. To do this, place the compass at X and Z, draw arcs that intersect above D. Draw a line from D through this intersection. This is the $90^\circ$ line.
• Now, bisect the angle between the $90^\circ$ line and the $120^\circ$ line (line DZ). Place the compass at the intersection of the $90^\circ$ line and the semicircle, and at Z. Draw arcs that intersect. Draw a line from D through this new intersection. This line forms an angle of $90^\circ + 15^\circ = 105^\circ$ with DC.

Along the $105^\circ$ line from D, use your compass to measure $6 \text{ cm}$. Place the compass point at D and draw an arc to intersect the $105^\circ$ line. Label this intersection point A. So, $AD = 6 \text{ cm}$.

• Draw a transversal line from A to any point on DC (e.g., draw a line from A to D, which is already done).
• Place the compass at D and draw an arc that intersects both DA and DC.
• With the same compass opening, place the compass at A and draw an arc that intersects DA.
• Measure the distance between the two intersection points on the arc centered at D.
• Transfer this distance to the arc centered at A, from the intersection point on DA. This marks a new point.
• Draw a straight line through A and this new point. This line is parallel to DC.

Along the parallel line from A, use your compass to measure $5 \text{ cm}$. Place the compass point at A and draw an arc to intersect the parallel line. Label this intersection point B. So, $AB = 5 \text{ cm}$.

Join point B to point C with a straight line. ABCD is the constructed trapezium.

Drawing the perpendicular and measuring the height

• Extend the line segment DC if necessary.
• Place the compass at B and draw an arc that intersects the line DC (or its extension) at two points, say P and Q.
• Place the compass at P and draw an arc below DC.
• With the same compass opening, place the compass at Q and draw another arc below DC, intersecting the previous arc.
• Draw a straight line from B through the intersection point of these two arcs. This line is perpendicular to DC. Label the intersection point on DC as H. The segment BH is the height of the trapezium.

Using your ruler, carefully measure the length of the segment BH.

The measured height of the trapezium will depend on the accuracy of your construction.