A trapezoid is a geometric shape with four corners. When constructing a trapezoid, it is important to consider that the two opposite sides are parallel, and the other two, on the contrary, are not parallel relative to each other. This word came to modernity from Ancient Greece and sounded like "trapezion", which meant "table", "dining table".
This article talks about the properties of a trapezoid described around a circle. We will also consider the types and elements of this figure.
Elements, types and signs of a geometric figure trapezoid
Parallel sides in this figure are called bases, and those that are not parallel are called sides. Provided that the sides are the same length, the trapezoid is considered isosceles. A trapezoid whose sides lie perpendicular to the base at an angle of 90 ° is called rectangular.
This seemingly straightforward figure has a considerable number of properties inherent in it, emphasizing its features:
- If you draw the middle line on the sides, then it will be parallel to the bases. This segment will be equal to 1/2 the difference of the bases.
- When constructing a bisector from any angle of the trapezoid, an equilateral triangle is formed.
- From the properties of the trapezoid described around the circumference, it is known that the sum of the parallel lateral sides must be equal to the sum of the bases.
- When constructing diagonal segments, where one of the sides is the base of the trapezoid, the resulting triangles will be similar.
- When constructing diagonal segments, where one of the sides is lateral, the resulting triangles will have an equal area.
- If we continue the side lines and build a segment from the center of the base, then the angle formed will be 90 °. The segment connecting the bases will be equal to 1/2 of their difference.
Properties of a trapezoid circled around a circle
To enclose a circle in a trapezoid is possible only under one condition. This condition is that the sum of the sides should be equal to the sum of the bases. For example, when constructing a trapezoid AFDM, AF + DM = FD + AM is applicable. Only then can a circle be enclosed in a trapezoid.
So, more about the properties of the trapezoid described around the circle:
- If a circle is enclosed in the trapezoid, then in order to find the length of its line crossing the figure in half, it is necessary to find 1/2 of the sum of the lengths of the sides.
- When constructing a trapezoid described around a circle, the formed hypotenuse is identical to the radius of the circle, and the height of the trapezoid in combination is also the diameter of the circle.
- Another property of an isosceles trapezoid described around a circle is that its lateral side is immediately visible from the center of the circle at an angle of 90 °.
A little more about the properties of a trapezoid enclosed in a circle
Only an isosceles trapezoid can be inscribed in a circle. This means that it is necessary to observe the conditions under which the constructed trapezoid AFDM will meet the following requirements: AF + DM = FD + MA.
Ptolemy’s theorem says that in a trapezoid enclosed in a circle, the product of the diagonals is identical and equal to the sum of the multiplied opposite sides. This means that when constructing the circle described near the trapezoid AFDM, it is applicable: AD × FM = AF × DM + FD × AM.
At school exams, tasks that require solving problems with a trapezoid are quite common. A large number of theorems must be memorized, but if you do not immediately learn it, it does not matter. It is best to periodically resort to a hint in textbooks, so that this knowledge by itself, without much difficulty fit into your head.