Q1.
Construct triangles having the following sidelengths (all the units are in cm): (a) 4, 4, 6 (b) 3, 4, 5 (c) 1, 5, 5 (d) 4, 6, 8 (e) 3.5, 3.5, 3.5
Answer
In every case draw the longest side as the base, then cut two arcs of the other two lengths.
| Set | Base to draw | Arcs from the two ends | Type of triangle |
|---|---|---|---|
| (a) 4, 4, 6 | 6 cm | 4 cm and 4 cm | Isosceles |
| (b) 3, 4, 5 | 5 cm | 3 cm and 4 cm | Scalene (right-angled) |
| (c) 1, 5, 5 | 5 cm | 1 cm and 5 cm | Isosceles (very thin) |
| (d) 4, 6, 8 | 8 cm | 4 cm and 6 cm | Scalene |
| (e) 3.5, 3.5, 3.5 | 3.5 cm | 3.5 cm and 3.5 cm | Equilateral |
Every set works, because in each one the longest length is smaller than the sum of the other two:
(a) 6 < 4 + 4 = 8 ✓
(b) 5 < 3 + 4 = 7 ✓
(c) 5 < 1 + 5 = 6 ✓
(d) 8 < 4 + 6 = 10 ✓
(e) 3.5 < 3.5 + 3.5 = 7 ✓
(b) 5 < 3 + 4 = 7 ✓
(c) 5 < 1 + 5 = 6 ✓
(d) 8 < 4 + 6 = 10 ✓
(e) 3.5 < 3.5 + 3.5 = 7 ✓
Why it happens: The two arcs meet only if together they can "reach across" the base. In (c) the reach is 1 + 5 = 6 against a base of 5 — just one centimetre to spare — so the triangle is long and thin, with a very small angle opposite the 1 cm side.
Check it yourself: Measure the angles of (b). You will get 37°, 53° and 90° — a right-angled triangle.