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Chapter 13: Problem 1

The vertices of quadrilateral ZEUS lie on the circle. What kind of quadrilateral is ZEUS?

Short Answer

Expert verified
ZEUS is a cyclic quadrilateral.

Step by step solution

01

Understanding Quadrilateral Properties

First, we need to understand the properties of different types of quadrilaterals. A quadrilateral whose vertices lie on a circle is called a *cyclic quadrilateral*. In cyclic quadrilaterals, the opposite angles add up to 180 degrees.
02

Identifying the Given Shape

We have a quadrilateral named ZEUS. According to the problem, ZEUS has all its vertices on a circle. This means it fits the definition of a cyclic quadrilateral.
03

Concluding the Quadrilateral Type

Since all vertices of quadrilateral ZEUS lie on the circle, by definition, it is a cyclic quadrilateral. The key property is that the opposite angles of ZEUS sum to 180 degrees, confirming its cyclic nature.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadrilateral Properties
A quadrilateral is a polygon with four sides and four angles. Understanding the varieties of quadrilaterals involves knowing their unique properties and characteristics. Some common types include:
  • Square: All sides are equal, and all angles are 90 degrees.
  • Rectangle: Opposite sides are equal, and all angles are 90 degrees.
  • Parallelogram: Opposite sides and angles are equal.
  • Trapezoid: Has only one pair of parallel sides.
  • Rhombus: All sides are equal, but angles are not 90 degrees.

When the vertices of a quadrilateral lie on a single circle, it is known as a cyclic quadrilateral. This type of quadrilateral has distinct properties, particularly regarding its angles, which sets it apart from others.
Opposite Angles
In any quadrilateral, we can find pairs of opposite angles. These angles are positioned across from one another within the four-sided figure.

In a cyclic quadrilateral, a special property exists: the sum of each pair of opposite angles is always 180 degrees.
  • This property can be expressed mathematically as: \[\angle A + \angle C = 180^\circ\]\[\angle B + \angle D = 180^\circ\]where \( A \), \( B \), \( C \), and \( D \) are the angles of the quadrilateral.
  • This unique property helps in verifying whether a given quadrilateral is cyclic or not.

It shows a balance in the quadrilateral's angular measurements, manifesting a distinctive cyclic property.
Circumscribed Circle
A circumscribed circle, also known as a circumcircle, is a circle that passes through all the vertices of a polygon, specifically in our case, a quadrilateral.

Not every quadrilateral can have a circumscribed circle. Only cyclic quadrilaterals, like the one in the exercise, fit this description.
  • A cyclic quadrilateral has its vertices touching the circle's circumference, confirming its "cyclic" quality.
  • The center of this circle is called the circumcenter, and it is equidistant from all vertices of the quadrilateral.

Studying the properties of a circumscribed circle can reveal additional insights about the nature of the quadrilateral that lies within it. This concept helps in solving geometric problems concerning quadrilaterals and circles.

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Most popular questions from this chapter

Quadrilateral ECHO is a rhombus in which \(\angle \mathrm{E}=112^{\circ}\) Is quadrilateral ECHO cyclic? (GRAPH CANT COPY)

Construct a circle having \(b\) as its radius and inscribe a rectangle having \(b\) as one of its sides in the circle.

Can the incenter of a triangle be outside of the triangle?

In some of the following exercises, two figures of different shapes can be constructed using the given parts. For example, in constructing a triangle having \(x\) and \(y\) as two of its sides and \(z\) as the altitude to the third side, we have a choice at the end. (IMAGE CAN'T COPY). Construct a triangle having \(d\) and \(f\) as two of its sides and \(c\) as the altitude to side \(f .\) (There are two possibilities.)

Draw and label segments having the following lengths for use in the exercises of this lesson: \(a, 1 \mathrm{cm} ; b, 2 \mathrm{cm} ; c, 3 \mathrm{cm} ; d, 4 \mathrm{cm}\) \(e, 5 \mathrm{cm} ;\) and \(f, 6 \mathrm{cm} .\) Before doing each construction, make a sketch of the completed figure and then try to develop a plan based on relations that you notice among its parts. Construct a right triangle having \(c\) and \(e\) as its legs.
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