Trapezium Parallel Sides Distance Calculation

The question asks for the shortest distance between the parallel sides PQ and SR of a trapezium PQRS. This distance is the height ($h$) of the trapezium relative to these parallel bases.

Given values are:

• Trapezium vertices: P, Q, R, S (in anticlockwise order)
• Parallel sides: PQ || SR
• Side lengths: PQ = 11 cm, QR = 4 cm, RS = 6 cm, SP = 3 cm

Geometric Setup for Trapezium Height

Let PQ be the longer base and SR be the shorter base. To find the height $h$, we can draw perpendiculars from the endpoints of the shorter base (S and R) to the longer base (PQ).

Let X and Y be the points on PQ such that SX is perpendicular to PQ and RY is perpendicular to PQ.

This forms:

• A rectangle between the perpendiculars, implying $XY = SR = 6$ cm.
• Two right-angled triangles: $\triangle SPX$ and $\triangle RQY$.
• The height of the trapezium is $h = SX = RY$.

The length of the base PQ can be expressed as $PX + XY + YQ$. Substituting the known values: $11 \text{ cm} = PX + 6 \text{ cm} + YQ$. This simplifies to $PX + YQ = 11 - 6 = 5 \text{ cm}$.

Pythagorean Theorem Application

Using the Pythagorean theorem on the two right-angled triangles:

• In $\triangle SPX$: $SP^2 = PX^2 + SX^2$
  $\implies 3^2 = PX^2 + h^2$
  $\implies 9 = PX^2 + h^2$ (Equation 1)
• In $\triangle RQY$: $QR^2 = RY^2 + YQ^2$
  $\implies 4^2 = h^2 + YQ^2$
  $\implies 16 = h^2 + YQ^2$ (Equation 2)

Solving for Trapezium Height

From Equation 1, $PX = \sqrt{9 - h^2}$. From Equation 2, $YQ = \sqrt{16 - h^2}$.

Substitute these expressions for PX and YQ into the equation $PX + YQ = 5$: $\sqrt{9 - h^2} + \sqrt{16 - h^2} = 5$.

This equation can be solved for $h$. An efficient method is to test the given options. Let's test $h = 2.4$ cm.

• Calculate PX: $PX = \sqrt{9 - (2.4)^2} = \sqrt{9 - 5.76} = \sqrt{3.24} = 1.8$ cm.
• Calculate YQ: $YQ = \sqrt{16 - (2.4)^2} = \sqrt{16 - 5.76} = \sqrt{10.24} = 3.2$ cm.
• Verify the sum: $PX + YQ = 1.8 \text{ cm} + 3.2 \text{ cm} = 5.0 \text{ cm}$.

The sum $PX + YQ = 5$ cm matches the condition derived from the base length PQ. Therefore, the height $h = 2.4$ cm is correct.

Final Answer Determination

The shortest distance between the parallel sides PQ and SR is the calculated height $h$.

The shortest distance is 2.4 cm.