# 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

- Construct 80719

Construct a rectangle ABCD if a = 8cm and the length of the diagonal AC is 13cm. Measure the length of the sides of the rectangle. - Right-angled 78394

A right-angled triangle was inscribed in a circle with a diameter of 20 cm, whose hypotenuse is the circle's diameter and has the largest possible area. Calculate the area of this triangle. - Triangle 73464

The given line is a BC length of 6 cm. Construct a triangle so that the BAC angle is 50° and the height to the side is 5.5 cm. Thank you very much. - Triangle 67504

Sestroj triangle HOP, if o = 6 cm, h = 8 cm and | PHO | = 90 °

- Quadrilateral 67384

Construct a quadrilateral ABCD if AB = 10cm, AD = 6cm, DC = 6.5cm and angle BCD = 90 degrees - Hypotenuse 65744

Construct a right triangle ABC with the hypotenuse AB: a) | AB | = 72 mm, | BC | = 51 mm b) | AB | = 58 mm, | AC | = 42 mm - Lunes of Hippocrates

Calculate the sum of the area of the so-called Hippocratic lunas, which were cut above the legs of a right triangle (a = 6cm, b = 8cm). Instructions: First, calculate the area of the semicircles above all sides of the ABC triangle. Compare the sum of the - Draw triangle

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

- MIT 1869

You know the length of hypotenuse parts 9 and 16, at which the hypotenuse of a right triangle is divided by a height. The task is to find the lengths of the sides of the triangle and the length of line x. This assignment was part of the Massachusetts Inst - Construct

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

Construct a square that has the same area as a rhombus ABCD if |AB| = 5cm, |AD| = 4cm and angle |DAB| = 30°. - 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? * - 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?

- Calculate 16223

The following elements are known in the right triangle ABC: a = 10 cm, height to side c h = 9.23 cm. Calculate o, R (radius of the inscribed circle), r (radius of the inscribed circle). - Construct rhombus

Construct rhombus ABCD if given diagonal length | AC | = 8cm, inscribed circle radius r = 1.5cm - Parallels and one secant

There are two different parallel lines, a, b, and line c, that intersect the two parallel lines. Draw a circle that touches all lines at the same time. - Construct 11511

Construct the diamond ABCD so that its diagonal BD is 8 cm and the distance of apex B from the line AD is 5 cm. Specify all options - Belongs 8412

Given a circle k(O; 2.5 cm), a line p: /Op/=4 cm, a point T: T belongs to p and at the same time /OT/=4.5 cm. We must find all the circles that will touch the circle k and the line p at point T.

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