# Thales' theorem - examples

1. Rectangle In rectangle with sides 10 and 8 mark the diagonal. What is the probability that a randomly selected point within the rectangle is closer to the diagonal than to any side of the rectangle?
2. OK circle Calculate the radius (circumradius) of the circle described right triangle with hypotenuse long 33 and one cathetus long 17.
3. The chord Side of the triangle inscribed in a circle is a chord passing through circle center. What size are the internal angles of a triangle, if one of them is 40°?
4. Height Is right that in any right triangle height is less or equal half of the hypotenuse?
5. The bridge Across the circle lakepasses through its center bridge over the lake. At three different locations on the lake shore are three fishermen A, B, C. Which of fishermen see the bridge under the largest angle?
6. Circumscribing Determine the radius of the circumscribed circle to the right triangle with legs 9 cm and 6 cm.
7. Circumferential angle Vertices of the triangle ΔABC lies on circle and divided it into arcs in the ratio 2:2:9. Determine the size of the angles of the triangle ΔABC.
8. Right triangle Draw a right triangle ABC if |AB| = 5 cm |BC| = 3 cm, |AC| = 4 cm. Draw Thales circle above the hypotenuse of the triangle ABC.
9. Semicircle In the semicircle with center S and the diameter AB is constructed equilateral triangle SBC. What is the magnitude of the angle ∠SAC?
10. Complete construction Construct triangle ABC if hypotenuse c = 7 cm and angle ABC = 30 degrees. / Use Thales' theorem - circle /. Measure and write down the length of legs.
11. Construct 1 Construct a triangle ABC, a = 7 cm, b = 9 cm with right angle at C, construct the axis of all three sides. Measure the length of side c (and write).
12. Circle tangent It is given to a circle with the center S and radius 3.5 cm. Distance from the center to line p is 6 cm. Construct a circle tangent n which is perpendicular to the line p.
13. Inscribed triangle To a circle is inscribed triangle so that the it's vertexes divide circle into 3 arcs. The length of the arcs are in the ratio 2:3:7. Determine the interior angles of a triangle.
14. Triangle SSA Construct a triangle ABC if |AB| = 5cm va = 3cm, CAB = 50 °. It is to create the analysis and construction steps.
15. Tangents construct Circle is given k (S; 2.5 cm) and an outer line p. Construct a tangent t of the circle that has with a line p angle 60°. How many solutions has the task?
16. Hypotenuse - RT A triangle has a hypotenuse of 55 and an altitude to the hypotenuse of 33. What is the area of the triangle?
17. Diagonals at right angle In the trapezoid ABCD this is given: AB=12cm CD=4cm And diagonals crossed under a right angle. What is the area of this trapezoid ABCD?
18. Circle described The radius of the circle described to the right triangle with 6 cm long leg is 5 cm. Calculate the circumference of this triangle.
19. Same area There is a given triangle. Construct a square of the same area.

Do you have an interesting mathematical example that you can't solve it? Enter it, and we can try to solve it.

To this e-mail address, we will reply solution; solved examples are also published here. Please enter e-mail correctly and check whether you don't have a full mailbox.