explain the cyclic quardilateral theorem
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Step-by-step explanation: The ratio between the diagonals and the sides can be defined and is known as Cyclic quadrilateral theorem. If there's a quadrilateral which is inscribed in a circle, then the product of the diagonals is equal to the sum of the product of its two pairs of opposite sides.
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Step-by-step explanation:
Cyclic Quadrilateral:
Definition
The word cyclic often means circular, just think of those two circular wheels on your bicycle. Quadrilateral means four-sided figure. Put them together, and we get the definition for cyclic quadrilateral: any four-sided figure (quadrilateral) whose four vertices (corners) lie on a circle.
Not every quadrilateral is cyclic, but I bet you can name a few familiar ones. Every rectangle, including the special case of a square, is a cyclic quadrilateral because a circle can be drawn around it touching all four vertices. However, no non-rectangular parallelogram is cyclic. There's no way to draw a circle around one that touches all four of the non-rectangular parallelogram's vertices.
Properties of Cyclic Quadrilaterals
There are more to cyclic quadrilaterals than circles. Here's a property of cyclic quadrilaterals that you'll soon see can help identify them:
The sum of opposite angles of a cyclic quadrilateral is 180 degrees.
In other words, angle A + angle C = 180, and angle B + angle D = 180.
There are many ways to prove this property, but the quickest one has to do with arc measures and inscribed angles. To refresh your memory, an inscribed angle is an angle that has its vertex on the circle's circumference. You can see that all the angles of our cyclic quadrilateral are inscribed angles. We also know the measure of an inscribed angle is half the measure of its intercepted arc, from the interior angle theorem. Let's use this to prove that the sum of opposite angles of cyclic quadrilateral is 180 degrees.
In our figure, the arc BCD intercepted by angle A and the arc DAB intercepted by angle C together make up the entire circle. So, the measures of arcs BCD and DAB together add up to 360 degrees. Remember that every circle has 360 degrees.
We know the cyclic quadrilateral's opposite angles A and C are inscribed angles. From the inscribed angle theorem, we also know that the measure of angle A is half the measure of its arc BCD, and the measure of angle C is half the measure of its arc DAB.