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Problem-solving Pack
MathematicsYear 8 · 8.14 Triangles
Problem-solving Pack
Name: _________________________________
Date: _________________ Class: ___________
These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.
1Problem 1 of 12
Triangle inequality - feasibility. A student wants to construct a triangle with whole-number side lengths and perimeter 12 cm.
(a) Use the triangle inequality to find all valid sets of three whole-number side lengths.
(b) Which is the only equilateral solution?
(c) Which sets give isosceles triangles?
(d) Which set is the special 3-4-5 right-angle triple?
(a) Use the triangle inequality to find all valid sets of three whole-number side lengths.
(b) Which is the only equilateral solution?
(c) Which sets give isosceles triangles?
(d) Which set is the special 3-4-5 right-angle triple?
Working space
2Problem 2 of 12
Constructing an isosceles triangle. Construct an isosceles triangle with base 6 cm and equal sides 5 cm.
(a) Describe the construction with compasses and straight-edge.
(b) Calculate the triangle's height using Pythagoras.
(c) Find its area.
(a) Describe the construction with compasses and straight-edge.
(b) Calculate the triangle's height using Pythagoras.
(c) Find its area.
Working space
3Problem 3 of 12
Algebra in a triangle. A triangle has angles , , .
(a) Set up and solve.
(b) Find the angles.
(c) Classify the triangle by angles.
(a) Set up and solve.
(b) Find the angles.
(c) Classify the triangle by angles.
Working space
4Problem 4 of 12
Pythagorean triples. A right-angled triangle has legs and , hypotenuse .
(a) Verify that , , are Pythagorean triples.
(b) Generate a new triple by multiplying by 4.
(c) Show that is a triple.
(a) Verify that , , are Pythagorean triples.
(b) Generate a new triple by multiplying by 4.
(c) Show that is a triple.
Working space
5Problem 5 of 12
Distance and triangle area. A triangle has vertices , , .
(a) Find each side.
(b) Find the perimeter.
(c) Find the area.
(d) Classify the triangle.
(a) Find each side.
(b) Find the perimeter.
(c) Find the area.
(d) Classify the triangle.
Working space
6Problem 6 of 12
Building uniqueness. Two triangles are constructed with the data:
A: sides 6, 8, 10.
B: sides 6, 8 with included angle 90°.
C: sides 6, 8 with included angle 45°.
(a) Are A and B congruent?
(b) Are A and C congruent?
(c) State which uniqueness rules apply.
A: sides 6, 8, 10.
B: sides 6, 8 with included angle 90°.
C: sides 6, 8 with included angle 45°.
(a) Are A and B congruent?
(b) Are A and C congruent?
(c) State which uniqueness rules apply.
Working space
7Problem 7 of 12
Triangle classification. Classify each triangle by sides AND angles:
(a) Sides 5, 5, 5
(b) Sides 3, 4, 5
(c) Sides 6, 8, 11
(d) Sides 7, 7, 10
(a) Sides 5, 5, 5
(b) Sides 3, 4, 5
(c) Sides 6, 8, 11
(d) Sides 7, 7, 10
Working space
8Problem 8 of 12
Find the missing side. A right-angled triangle has hypotenuse 17 and one leg 8. Find the other leg.
Use the result to find a value of in the triangle with sides , , — assume this is right-angled.
Use the result to find a value of in the triangle with sides , , — assume this is right-angled.
Working space
9Problem 9 of 12
Angles in an isosceles triangle. An isosceles triangle has equal sides of length 13 cm and base 10 cm.
(a) Find the perpendicular height from apex to base.
(b) Find the area.
(c) Find the two base angles (using inverse tan).
(d) Find the apex angle.
(a) Find the perpendicular height from apex to base.
(b) Find the area.
(c) Find the two base angles (using inverse tan).
(d) Find the apex angle.
Working space
10Problem 10 of 12
Outdoors application. A 6 m vertical pole stands on horizontal ground. A wire is anchored from the top of the pole to a point on the ground 8 m from the base of the pole.
(a) Sketch the triangle formed.
(b) Find the length of the wire.
(c) Find the angle the wire makes with the ground.
(a) Sketch the triangle formed.
(b) Find the length of the wire.
(c) Find the angle the wire makes with the ground.
Working space
11Problem 11 of 12
Triangle similarity. Two triangles ABC and DEF are similar with linear scale factor .
(a) The smaller has sides 6, 8, 10. Find the sides of the larger.
(b) Find the ratio of perimeters.
(c) Find the ratio of areas.
(a) The smaller has sides 6, 8, 10. Find the sides of the larger.
(b) Find the ratio of perimeters.
(c) Find the ratio of areas.
Working space
12Problem 12 of 12
Investigating triangle types. For each set of angles, decide if the triangle is acute / right / obtuse, and explain.
(a) 30°, 60°, 90°
(b) 50°, 60°, 70°
(c) 100°, 40°, 40°
(d) 89°, 89°, 2°
(a) 30°, 60°, 90°
(b) 50°, 60°, 70°
(c) 100°, 40°, 40°
(d) 89°, 89°, 2°
Working space