AC9M7SP02 • Homework
Classifying Triangles, Quadrilaterals and Polygons
Use defining side, angle and parallel-line properties. Show reasons rather than classifying from appearance alone.
Part A — Short response
- Why is a triangle made from fixed-length hinged strips rigid while a quadrilateral can flex? (2)
- Can 4 cm, 7 cm and 12 cm form a triangle? Justify using triangle inequality. (2)
- Classify a triangle with side lengths 6 cm, 6 cm and 9 cm by sides. (2)
- A quadrilateral has both pairs of opposite sides parallel. Name the most general special quadrilateral guaranteed. (2)
- Classify a triangle with angles 110°, 35° and 35° by both angles and sides. (2)
Part B — Extended response
- For each set, decide whether a triangle is possible and classify it by sides if possible: (a) 5,5,8 cm; (b) 3,4,9 cm; (c) 7,8,10 cm. Explain each decision. (5)
- Explain why engineers add diagonal braces to rectangular or parallelogram frames. Use triangle rigidity in your explanation. (5)
- Classify: A = 60°,60°,60°; B = 45°,45°,90°; C = 120°,30°,30°. Give both side and angle classifications and justify. (5)
- Explain the hierarchy connecting squares, rectangles, rhombuses and parallelograms. Include at least three true “every … is …” statements and one false converse. (5)
- A quadrilateral has two pairs of adjacent equal sides and one line of symmetry. Identify the likely shape, state the defining evidence, and explain why orientation on the page should not affect classification. (5)
Part C — Enrichment
11. Stable-frame design. A designer can build a frame as a triangle, square, trapezium or parallelogram using straight rods and hinged vertices. Compare which frames are inherently rigid and which can flex. Explain how adding diagonals changes the flexible frames, connect your reasoning to triangle inequality and triangle construction, and give a real structural application such as a bridge or roof truss.
Answers
1
Three fixed sides determine a triangle's shape. Four hinged sides do not fix all quadrilateral angles, so the frame can shear.
2
No. 4 + 7 = 11, which is less than 12.
3
Isosceles: two sides are equal.
4
Parallelogram.
5
Obtuse isosceles: one angle is greater than 90° and two angles are equal, so the opposite sides are equal.
6
5,5,8: possible, isosceles. 3,4,9: impossible because 3+4<9. 7,8,10: possible, scalene.
7
A diagonal divides the quadrilateral into two triangles. Fixed triangle side lengths constrain the angles, preventing the frame from shearing.
8
A: equilateral acute. B: isosceles right-angled. C: isosceles obtuse.
9
Every square is a rectangle and rhombus; every rectangle is a parallelogram; every rhombus is a parallelogram. A false converse is “every rectangle is a square”.
10
Kite is supported by two pairs of adjacent equal sides and typical symmetry. Rotation does not alter side lengths, angles, parallelism or equality.
11
Triangles are inherently rigid; unbraced quadrilaterals can flex. A diagonal triangulates a quadrilateral and locks its shape. Triangle inequality ensures three rods can meet to form a non-degenerate triangle. Applications include bridge trusses, roof frames and towers.