Chapter 3.4 — Exercise 3.3 — SSSDD Construction

The General Method

For a quadrilateral where two diagonals and three of the four sides are known, only one side has to remain unlabelled at the start — it turns out as a byproduct of the construction rather than being drawn directly from a measurement:

• Draw one known side connecting two vertices that are each an endpoint of one diagonal.
• Swing an arc from each end of that side, using the two remaining known sides as radii, to locate the third vertex of this first triangle.
• Join both ends of the base to that third vertex, forming the first triangle.
• Swing two more arcs — one from each of two already-placed vertices, using the two given diagonal lengths as radii — to locate the fourth vertex.
• Join the fourth vertex to the appropriate two vertices to complete the quadrilateral, including drawing in the one side that was never given a length directly.

Two Diagonals Remove the Hinge Twice Over

A four-sided figure with only its side lengths fixed can still flex, like a hinge, into more than one shape — a single diagonal removes that flex once, by fixing the distance across one pair of opposite corners. Giving a second diagonal instead of a fourth side does the same job from a different angle: rather than fixing one more side, it fixes the distance across the other pair of opposite corners. Between three sides and two diagonals, every possible way the quadrilateral could have flexed is accounted for, which is exactly why five measurements — no more, no fewer — are needed here too, matching every other method in this chapter, whether the five measurements come as sides and diagonals, sides and angles, or some other combination entirely.

Worked Example — Quadrilateral GOLD

Construct quadrilateral GOLD with OL = 7.5 cm, GL = 6 cm, LD = 5 cm, DG = 5.5 cm and diagonal OD = 10 cm. Here OL, LD and DG are three of the four sides (only GO is not directly given), and GL and OD are the two diagonals.

1. Draw line segment OL of length 7.5 cm.
2. With L as centre, draw an arc of radius 5 cm (side LD).
3. With O as centre, draw an arc of radius 10 cm (diagonal OD), cutting the previous arc at D.
4. Join L–D and O–D to complete triangle OLD.
5. With L as centre, draw an arc of radius 6 cm (diagonal GL).
6. With D as centre, draw an arc of radius 5.5 cm (side DG), cutting the previous arc at G.
7. Join G–L, G–O and G–D to complete quadrilateral GOLD.

A Second Worked Example — Quadrilateral PQRS

Construct quadrilateral PQRS with PQ = 4.2 cm, QR = 3 cm, PS = 2.8 cm, and diagonals PR = 4.5 cm and QS = 5 cm.

1. Draw line segment PQ of length 4.2 cm.
2. With Q as centre, draw an arc of radius 5 cm (diagonal QS).
3. With P as centre, draw an arc of radius 2.8 cm (side PS), cutting the previous arc at S.
4. Join P–S and Q–S to complete triangle PQS.
5. With Q as centre, draw an arc of radius 3 cm (side QR).
6. With P as centre, draw an arc of radius 4.5 cm (diagonal PR), cutting the previous arc at R.
7. Join Q–R, P–R and S–R to complete quadrilateral PQRS.

Notice the order differs slightly from GOLD's construction — here the first triangle (PQS) is built using one side and one diagonal from a shared base, then the second triangle uses the remaining side and diagonal from a different shared vertex. The underlying principle stays the same: any three known lengths that share two vertices between them can define a triangle, and two such overlapping triangles together fix the whole quadrilateral.

Counting the Given Measurements Carefully

SSSDD asks for three sides and two diagonals — five measurements in total, matching every other method in this chapter — but it's easy to miscount when a problem lists them in a jumbled order rather than neatly grouped. Before starting either worked example above, it helps to sort the given values into two short lists side by side: which lengths connect adjacent vertices (sides), and which connect opposite vertices (diagonals). GOLD's five measurements split into three sides (OL, LD, DG) and two diagonals (GL, OD); doing this sorting explicitly, rather than working it out on the fly while drawing, is what prevents an arc being drawn with the wrong radius.

Identifying Which Two Vertices to Start From

The trickiest part of this exercise is deciding which known side to draw first, since with two diagonals in the mix there's more than one valid starting pair. A reliable approach: pick the side whose two endpoints are also each connected to one of the two given diagonals — that guarantees the first triangle you build uses only measurements you already have, with nothing left to guess.

Two Diagonals Give More Certainty Than One

Compared with Exercise 3.2's single-diagonal method, having two diagonals here means both triangles making up the quadrilateral are pinned down by a diagonal directly, rather than one triangle borrowing its closing side from the first. In GOLD, triangle OLD uses diagonal OD as one of its own sides, and the second half of the construction uses diagonal GL the same way for the triangle involving G. This gives SSSDD a kind of built-in symmetry that SSSSD doesn't have — neither triangle depends on "inheriting" a side from the other, since each has its own diagonal to complete it. In practice this also makes SSSDD slightly more forgiving to verify afterward: each half of the quadrilateral can be checked as its own complete SSS triangle, independent of whether the other half was drawn accurately.

Easy Ways to Mix Up a Side and a Diagonal

• Trying to draw the unlabelled fourth side first. One side is never given directly in this construction — it's meant to emerge naturally once both triangles are built, not to be measured or estimated beforehand.
• Mixing up which measurement is a side and which is a diagonal. Since both are just numbers with units, keep a clear note of which given length connects adjacent vertices (a side) and which connects opposite vertices (a diagonal) before drawing any arc.
• Building the two triangles in the wrong order. Always construct the triangle that uses only already-known measurements first — attempting the second triangle before the first is complete leaves you without the vertex you need as an arc centre.

Shifting From Diagonals to Angles

Having built quadrilaterals from sides plus diagonals in this exercise and the last, Exercise 3.4 shifts entirely to angles, constructing a quadrilateral from two adjacent sides and three angles instead. For the single-diagonal version of this same triangle-splitting idea, revisit Exercise 3.2.