# Thales' theorem - math 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°.

#### Number of problems found: 24

• MIT 1869
You know the length of hypotenuse parts 9 and 16, at which the hypotenuse of a right triangle is divided by a perpendicular running from its opposite vertex. The task is to find the lengths of the sides of the triangle and the length of the line x. This a
• The amphitheater
The amphitheater has the shape of a semicircle, the spectators sit on the perimeter of the semicircle, 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?
• 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 a line c, that intersect the two parallel lines. Draw a circle that touches all lines at the same time.
• 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?
• 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.
• Two heights and a side
Construct triangle ABC when the given side is c = 7 cm, height to side a va = 5 cm and height to side b: vb = 4 cm.
• Same area
There is a given triangle. Construct a square of the same area.
• Hypotenuse - RT
A triangle has a hypotenuse of 55 and an altitude to the hypotenuse of 33. What is the area of the triangle?
• 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.
• 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.
• 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 with a line p angle 60°. How many solutions have the task?
• 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).
• Triangle SSA
Construct a triangle ABC if |AB| = 5cm va = 3cm, CAB = 50 °. It is to create the analysis and construction steps.
• 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.
• 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.
• The chord
The side of the triangle inscribed in a circle is a chord passing through the circle center. What size are the internal angles of a triangle if one of them is 40°?
• 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?
• Semicircle
In the semicircle with center S and the diameter AB is constructed equilateral triangle SBC. What is the magnitude of the angle ∠SAC?
• Circumferential angle
Vertices of the triangle ΔABC lay on the circle and divided into arcs in the ratio 7:8:7. Determine the size of the angles of the triangle ΔABC.

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