Inscribing a Quadrilateral

What does it mean to "inscribe" something in the world of mathematics? When you hear the term "inscribe," it essentially means that something is being drawn or marked. So when we talk about inscribing a quadrilateral in a circle, we're talking about drawing a quadrilateral inside of a circle. You could form the same figure by drawing a circle around (circumscribing) a quadrilateral. But why would we want to create this type of diagram in the first place? What can it teach us about math? Let's find out:

What does a quadrilateral inscribed in a circle look like?

A quadrilateral inscribed in a circle looks like this:

As we can see, each of the vertices of the quadrilateral must touch a part of the perfect circle. You might have also noticed that this is only possible under certain circumstances. We call these circles "cyclic" quadrilaterals.

What happens if we change the shape of this quadrilateral in such a way that the opposite angles are not supplementary?

Remember, angles are supplementary if they add up to a total of 180 degrees. In other words, they form a straight line.

That's right! A quadrilateral cannot be inscribed in a circle unless its opposite angles are supplementary. This is an important rule to remember for later math problems. We can write this as:

$\angle Q+\angle S=180°$

$\angle R+\angle P=180°$

Why does this make sense?

Proof: Sum of Opposite Angles of an Inscribed Quadrilateral is 180 Degrees.

To understand why the sum of opposite angles of an inscribed quadrilateral is 180 degrees, let's consider the following:

When a quadrilateral is inscribed in a circle, its vertices lie on the circle. The angle formed at each vertex of the inscribed quadrilateral is called an inscribed angle.

Recall that an inscribed angle of a circle is half of the central angle that subtends the same arc on the circle. Let's denote the inscribed angles by A,B,C, and D, and the corresponding central angles by α,β,γ, and δ.

Now, consider the following:

The sum of the central angles α and γ is equal to the total angle at the center, which is 360 degrees. Therefore, $\alpha +\gamma =360$ . By the same logic, $\beta +\delta =360$ .

Using the property of inscribed angles $(\mathrm{inscribed\; angle}=\frac{1}{2}\times \mathrm{central\; angle})$ , we get:

$A=\frac{1}{2}\times \alpha$

$C=\frac{1}{2}\times \gamma$

$B=\frac{1}{2}\times \beta$

$D=\frac{1}{2}\times \delta$

Now let's add A and C:

$A+C=\frac{1}{2}\times \alpha +\frac{1}{2}\times \gamma$

Recall that $\alpha +\gamma =360$ , so:

$A+C=\frac{1}{2}\cdot 360=180$

Similarly, for B and D:

$B+D=\frac{1}{2}\beta +\frac{1}{2}\delta$

Since $\beta +\delta =360$ :

$B+D=\frac{1}{2}\cdot 360=180$

Hence, the sum of opposite angles of an inscribed quadrilateral is 180 degrees. This proof shows that the circle theorem regarding inscribed quadrilaterals is indeed valid.

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