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# Triangle

1.Characterize a triangle

Solution:

2.Prove for the exterior angles of a triangle: α*+ β*+ γ* = 360°.

α + α* = 180°

β + β* = 180°

γ + γ* = 180°

_____________

α + β + γ + α* + β* + γ* = 540°

α* + β* + γ* = 540° – (α + β + γ)

α* + β* + γ* = 540° - 180°

α* + β* + γ* = 360°

3. A triangles‘ side size c = 10 cm. For it‘s angles stands: α:β:γ = 3:5:10.

a) perimeter

b) area

Perimeter

Area

4.An equilateral triangle‘s side length a = 10 cm.

5.John stands 5 m and 8 m from the edges of a tree. He can see the tree under the angle of 60°.

a = 5 m

b = 8 m

γ = 60°

x

^{2}= a

^{2}+ b

^{2}-2ab.cosγ

x

^{2}= 5

^{2}+ 8

^{2}-2.5.8.cos60°

x

^{2}= 25 + 64 – 80.0,5

x

^{2}= 25 + 64 – 40

x

^{2}= 49

x = 7

The tree is 7 m long.

6.There are three tangent circles with radius r_{1} = 2 cm, r_{2} = 3cm, r_{3} = 4cm. Centres of the circles define a triangle.

Inner angles of the triangle equal α = 44,41°, β = 78,46° and γ = 57,13°.

7.Two trains left a train station in the same time on two tracs forming the angle φ = 156°. Their speeds are v_{1} = 13 ms^{-1}, v_{2} = 14,5 ms^{-1}.

The distance between the trains after 5 minutes is 8.07 km.

8. Forces F_{1}, F_{2} act from the same point of application. Angle between the forces φ = 60°.

_{1}= 42N, F

_{2}= 35N.

Determine the angles between the forces F

_{1}, F

_{2}and the resultant power.

Solution:

The resultant power F = 61,87N.

The angles are α = 36° and β = 24°.

9.The angle between the coasts of a river observed from a 20 m height tower which is 20 m far from the river is 15°.

The width of the river is x = 14,64 m.

10. An object of weight m = 1563 kg hangs on a horizontal girder. It‘s fastened by two ropes of different length. The angles between a ropes and the girder equal α = 48°, β = 62°.

The forces acting on the ropes are F

_{1}= 12367,5 N and F

_{2}= 14697,9 N.