251. Between the Same Parallels · Same height, different shapes

Try to break it

Drag the top handle H to slide the upper parallel line — all three shapes change height together. Drag P, R, and T to slide the shapes sideways along the bottom line; the bases shift, but every shape's height stays locked to the same perpendicular distance between the parallels. Try to give any shape a different height — impossible.

How you build it

Place a triangle, a rectangle and a parallelogram between the same two parallels.

• Triangle tool: click two points on the lower line for the base, then a point on the upper line for the apex.
• Rectangle tool: click a bottom corner on the lower line, then the opposite top corner on the upper line. Same height as the triangle — the parallels are horizontal, so it fits exactly.
• Polygon tool: click two corners on the lower line, then two corners on the upper line (shifted sideways), then click the first corner again to close — a slanted parallelogram of the same height.

The proof, step by step

Prove that figures between the same parallels have the same height.

• Given two parallel lines AB and CD.
• Figures are placed between them with bases on CD and opposite vertices on AB.
• The perpendicular distance between AB and CD is constant everywhere.
• Therefore, all figures between the same parallels share the same altitude.

Worked example

In the figure, parallelogram PQRS, rectangle EFGH, and triangle LMN are between the same parallels AB and CD. If the distance between AB and CD is 12 cm, what is the altitude of triangle LMN?

By definition, figures between the same parallels share the same altitude. Since the distance between the parallels is 12 cm, the altitude of the triangle is also 12 cm.

• 6 cm
• 12 cm — correct
• 24 cm
• Cannot be determined