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Math Exercises & Math Problems: Triangles and Properties of a Triangle

 

 

 

 

1. An equilateral triangle has a side length of 4 cm. Find the perimeter, the area and the size of internal and external angles of the triangle.

 

2. The external angle of an isosceles triangle is 87°. Find the interior angles of the triangle.

 

3. One of legs of a right-angled triangle has a length of 12 cm. At what distance is a midpoint of the hypotenuse from the other leg ?

 

4. Consider a triangle ABC . What is the size of an angle between the altitude from side BC and the altitude from side AB, where the angles at vertices A, B are α = 30°, β = 45°?

 

5. In a right-angled triangle with the hypotenuse of 50 cm we know the perimeter of the triangle P = 1.2 m and the area A = 6 dm2. Find the length of the legs and the size of all internal angles of the triangle.

 

6. An isosceles triangle ABC has a base |AB| = 12 cm. An altitude hc from the base is 10 cm long. Find the length of the arm and the median of the arm.

 

7. The ladder 8.5 m long is leaning against a vertical wall. Its lower end rests on the ground at a distance of 1.8 m from the wall.
a) To what height of the wall reaches the upper end of the ladder ?
b) Under what angle is the ladder leaning against the wall ?

 

8. In a triangle MPN is |NP| = 7 cm, |PM| = 13 cm and the altitude from side MN is |PP'| = 5 cm. Find the length of side MN.

 

9. A perimeter of an isosceles triangle ABC is 60 cm long, the square of an altitude from the base hc2 = 60. Find the length of a base and the length of arms of the triangle.

 

10. In an isosceles triangle ABC the size of an angle at the vertex A is equal to 42°. At the arm AB is constructed such a point D, that |CB| = |CD|. Find an angle ACD.

 

11. Find the size of angles of the triangle ABC, if |α| = 2|β| and |β| = 3|γ|.

 

12. Three circles with radii r1 = 5 cm, r2 = 10 cm and r3 = 12 cm touch each other from the outside. Find the length of sides and the size of angles of the triangle that is formed by connecting the circle centers.

 

13. Find the size of angles and sides of the triangle, in which is valid for the size of angles α : β : γ = 3 : 4 : 5 and the side lying opposite to the angle α is of a length a = Square root of two.

 

14. On the top of the hill is 30 m high tower. We see its top and its bottom from a certain point in the valley below the elevation angles α, β. How high is the top of the hill above the horizontal position of our observation, if α = 28°30', β = 30°40' ?

 

15. From the 20 m high tower distant 20 m from the river appears width of the river at an angle of 15°. How wide is the river at this point ?

 

16. Find the size of angles and sides of the triangle ABC, if you know α = 51°19', β = 67°38' and the altitude from side c is hc = 28.

 

17. In a triangle ABC is an angle α lying opposite to the side a = Square root of three twice the size of an angle β lying opposite to the side b = 1. Find the perimeter and the area of the triangle ABC.

 

18. Find the length of the sides of the triangle ABC, in which α = 113°, β = 48° and the radius of the circumscribed circle of the triangle is r = 10 cm.

 

19. Find the side lengths and the sizes of the angles of the triangle ABC, which is given by a = 8.4, β = 105°35', median ma = 12.5.

 

20. Three circles with radii r1 = 5, r2 = 4 and r3 = 6 touch each other from the outside. Find the area of a pattern lying between them.

 

21. Find out if a triangle with side lengths a = 11, b = 14, c = 18 has an obtuse internal angle.

 

22. Find the side lengths and the sizes of the internal angles of the triangle ABC, which is given by A = 501.9, α = 15°28' and β = 45°.

 

23. Two forces of sizes 72 N and 58 N are acting at the same point of a solid body in a directions enclosing an angle 72°30'. Find the size of the resultant of the two forces.

 

24. Parallelogram ABCD has the area of 40 cm2, |AB| = 8.5 cm and |BC| = 5.65 cm. Find the lengths of its diagonals.

 

25. Find the length of all sides of a right triangle ABC, if you know the length of medians ma = 12 and mb = 15.

 

26. Verify whether a triangle whose sides have lengths 2, n – n–1, n + n–1 is right-angled.

 

27. The circle has inscribed and circumscribed square. The difference between the areas of the squares is 18. Find the radius of the circle.

 

28. Find the radius of the circle in which the chord of a circle distant from the center of the circle of 8 cm is 13 cm longer than the radius of the circle.

 

29. Rhombus has the area of A = 120 and the ratio of its diagonals is e : f = 5 : 12. Find the size of its side a, altitude h and diagonals e, f.

 

30. Consider a right-angled triangle ABC given by its leg a = 5 and the altitude hc = 3. Find the length of sides b and c.

 

 

 

 

 

 

You might be also interested in:

- Perimeter and Area of Plane Figures

- Volume and Surface Area of Solids

- Trigonometry

- Geometric Construction Exercises

 

 
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