Question

У прямокутну трапецію вписано коло. Точка дотику ділить більшу бічну сторону на відрізки завдовжки 8 см і 50 см. Знайти периметр даної трапеції, якщо радіус вписаного кола дорівнює 20 см. ДОПОМОЖІТЬ БУДЬ ЛАСКА

Answer 1

Відповідь:

Пояснення:

Answer 2

To find the perimeter of the trapezium, we need to determine the lengths of all its sides. Let's analyze the given information step by step.

Given Information:

- The trapezium is inscribed in a circle. - A point of tangency divides the longer side into two segments, measuring 8 cm and 50 cm. - The radius of the inscribed circle is 20 cm.

Approach:

1. Let's consider the trapezium ABCD, where AB is the longer base, CD is the shorter base, and AD and BC are the non-parallel sides. 2. We know that the point of tangency divides the longer side AB into two segments, measuring 8 cm and 50 cm. Let's denote the point of tangency as E. 3. Since the trapezium is inscribed in a circle, the tangents from a point to a circle are equal in length. Therefore, AE = BE = 8 cm and CE = DE = 50 cm. 4. Let's denote the center of the inscribed circle as O and the radius as r. In this case, r = 20 cm. 5. The diagonals of a trapezium bisect each other. Therefore, OE is the perpendicular bisector of AB. This means that OE is also the height of the trapezium. 6. We can use the Pythagorean theorem to find the length of OE. The right triangle OEB has OE as the hypotenuse and BE as one of the legs. The other leg is r, which is 20 cm. Using the Pythagorean theorem, we can find OE. - OE^2 = BE^2 + OB^2 - OE^2 = 8^2 + 20^2 - OE^2 = 64 + 400 - OE^2 = 464 - OE ≈ √464 ≈ 21.54 cm 7. Now, we have all the necessary lengths to calculate the perimeter of the trapezium. - Perimeter = AB + BC + CD + DA - Perimeter = AE + BE + BC + CD + DE + DA - Perimeter = 8 cm + 8 cm + BC + CD + 50 cm + 50 cm - Perimeter = 16 cm + BC + CD + 100 cm - Perimeter = 116 cm + BC + CD

To find the values of BC and CD, we need to use the properties of a trapezium inscribed in a circle. Unfortunately, I couldn't find specific information about this property in the search results. However, I can provide you with a general approach to solve this problem.

General Approach to Find BC and CD:

1. Let's consider the trapezium ABCD, where AB is the longer base, CD is the shorter base, and AD and BC are the non-parallel sides. 2. Since the trapezium is inscribed in a circle, we can use the properties of cyclic quadrilaterals to find the lengths of BC and CD. 3. One property states that the opposite angles of a cyclic quadrilateral are supplementary. In our case, angle BAD and angle BCD are opposite angles. 4. We can use the property of supplementary angles to find the measures of angle BAD and angle BCD. 5. Once we know the measures of these angles, we can use trigonometric ratios or other geometric properties to find the lengths of BC and CD.

I apologize for not being able to provide the specific values of BC and CD. However, I hope the general approach outlined above will help you solve the problem. If you have any further questions, please let me know!