Problems — 8.14 Triangles | Mathematics Department

1

**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?

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2

**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.

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3

**Algebra in a triangle.** A triangle has angles $(2x)°$, $(3x + 10)°$, $(x - 10)°$. (a) Set up and solve. (b) Find the angles. (c) Classify the triangle by angles.

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4

**Pythagorean triples.** A right-angled triangle has legs $a$ and $b$, hypotenuse $c$. (a) Verify that $(3, 4, 5)$, $(5, 12, 13)$, $(8, 15, 17)$ are Pythagorean triples. (b) Generate a new triple by multiplying $(3, 4, 5)$ by 4. (c) Show that $(7, 24, 25)$ is a triple.

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5

**Distance and triangle area.** A triangle has vertices $A(0, 0)$, $B(8, 0)$, $C(4, 6)$. (a) Find each side. (b) Find the perimeter. (c) Find the area. (d) Classify the triangle.

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6

**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.

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7

**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

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8

**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 $x$ in the triangle with sides $x$, $x + 1$, $x + 17$ — assume this is right-angled.

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9

**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.

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10

**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.

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11

**Triangle similarity.** Two triangles ABC and DEF are similar with linear scale factor $k = 1.5$. (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.

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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°

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