# Thales' theorem - practice problems

The Tales theorem says that if A, B, C are points on a circle, where AC is the diameter of the circle, then the angle ABC is the right angle. The Tales circle is the set of vertexes of right angles of right triangles constructed above the diameter of the circle. The Tales theorem results directly from the inscribed angle theorem. Simple proof - the radius joining point C divides a rectangular triangle into two isosceles triangles, and the sum of the angles in each triangle is 180°.Direction: Solve each problem carefully and show your solution in each item.

#### Number of problems found: 44

- Same area

There is a given triangle. Construct a square of the same area. - Height

Is it true that the height is less or equal to half of the hypotenuse in any right triangle? - OK circle

The right triangle has hypotenuse long 33 and one cathetus long 17. Calculate the radius (circumradius) of the circle described. - Semicircle

The semicircle with center S and the diameter AB is constructed equilateral triangle SBC. What is the magnitude of the angle ∠SAC? - 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. - Circumscribing

Find the radius of the circumscribed circle to the right triangle with legs 6 cm and 3 cm. - Draw triangle

Construct right triangle MNO with hypotenuse o = 5 cm and angle MNO = 37° - Triangle 45671

Draw a right triangle with side a = 5 cm, c = 8 cm. The right angle is at vertex C. What is the size of side b? * - Spectators 7562

The theater has the shape of a semicircle. A podium is the diameter of a semicircle. Spectators K, L, M, N, and O, sit around the perimeter. Who sees the podium at the greatest angle? - Distance 4527

There are two points, K and L, KL = 4 cm. Draw a line p passing through point K and having a distance of 4 cm from point L. - Construct rhombus

Construct rhombus ABCD if given diagonal length | AC | = 8cm, inscribed circle radius r = 1.5cm - Quadrilateral 67384

Construct a quadrilateral ABCD if AB = 10cm, AD = 6cm, DC = 6.5cm and angle BCD = 90 degrees - Tangent 6770

A circle k (S; 2.5 cm) and a point L are given if | SL | = 4cm. Make a tangent to the circle passing through point L. - Construct

Construct a rhombus ABCD with side a = 7cm, b = 5cm, whose diagonal e is perpendicular to side b. - The amphitheater

The amphitheater has the shape of a semicircle, the spectators sit on the perimeter of the semicircle, and the stage forms the diameter of the semicircle. Which of the spectators, P, Q, R, S, T, sees the stage at the greatest viewing angle? - Circle tangent

It is given to a circle with the center S and a radius of 3.5 cm. The distance from the center to line p is 6 cm. Construct a circle tangent n which is perpendicular to the line p. - Tangents construct

The circle k is given k (S; 2.5 cm) and an outer line p. Construct a tangent t of the circle that has a line p angle 60°. How many solutions have the task? - The chord

A chord passing through its center is the side of the triangle inscribed in a circle. What size are the internal angles of a triangle if one of them is 40°? - The bridge

Across the circle, the lake passes through its center bridge over the lake. At three different locations on the lakeshore are three fishermen, A, B, and C. Which of the fishermen see the bridge from the largest angle? - Touch circle

Point A has a distance (A, k) = 10 cm from a circle k with radius r = 4 cm and center S. Calculate: a) the distance of point A from the point of contact T if the tangent to the circle is drawn from point A b) the distance of the contact point T from the l

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