KJSEA 2025 Maths Q34 — Geometric Construction of a Parallelogram
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The Question
“A tabletop is a parallelogram ABCD in which AB is 6 cm, AD is 4 cm and angle DAB is 45 degrees. Using a ruler and a pair of compasses only, construct the parallelogram ABCD. Then drop a perpendicular from D to meet AB at E and measure the length of DE.”
Draw the base AB
Start with the side you know completely. Using a ruler, draw a straight line and mark off AB exactly 6 cm long. This fixed base gives you the two vertices A and B to build the rest of the shape around.
Construct the 45 degree angle at A
You cannot measure 45 degrees accurately with a protractor in a construction question, so you build it with compasses. First construct a 90 degree angle at A by erecting a perpendicular to AB, then bisect that right angle. Half of 90 degrees is exactly 45 degrees, which is the angle DAB you need.
Mark AD along the 45 degree line
Open your compasses to 4 cm. Place the point at A and cut an arc across the 45 degree line you just made. The point where the arc crosses is D, so that AD is exactly 4 cm and sits at 45 degrees to the base.
Complete the parallelogram
In a parallelogram opposite sides are equal and parallel. From D swing an arc of length 6 cm (equal to AB), and from B swing an arc of length 4 cm (equal to AD). Where the two arcs meet is C. Joining D to C and B to C gives DC parallel to AB and BC parallel to AD, completing ABCD.
Drop the perpendicular from D and measure DE
From D, construct a line straight down to the base AB so that it makes a right angle where it lands, calling that point E. This DE is the height of the parallelogram. Measuring it with a ruler gives about 2.8 cm.
DE is the perpendicular distance from D to AB, which is the height of the tabletop above its base.
Final Result
The perpendicular DE measures approximately 2.8 cm.
Why this method works
The perpendicular DE is the vertical height of the parallelogram, and it depends only on the slant side AD and the angle it makes with the base. If you picture the right-angled triangle ADE, AD is the hypotenuse and DE is the side opposite the 45 degree angle at A. By the definition of sine, the opposite side equals the hypotenuse times the sine of the angle, so DE equals AD times the sine of 45 degrees. That is why a careful construction and a direct calculation give the same value.
DE = AD \times \sin 45^\circ = 4 \times 0.71 \approx 2.8 \text{ cm}