In this unit, your student will study properties of circles. Students start by exploring new vocabulary. In prior units, students worked with radii and diameters of circles. Here, several new concepts are defined: Chords are line segments with endpoints on a circle. A line tangent to a circle intersects the circle at exactly one point. An arc is the portion of a circle’s circumference between two endpoints.

Next, students examine inscribed and circumscribed circles. A circle is circumscribed about a polygon if it passes through each of the polygon’s vertices, and a circle is referred to as an “inscribed circle” if it is tangent to all sides of the polygon.

All triangles have both circumscribed and inscribed circles. To draw a circumscribed circle for a triangle, construct the perpendicular bisectors of the triangle’s sides. These three lines meet at a point called the triangle’s circumcenter. A circle centered at this point, with radius set to the distance between the circumcenter and a vertex of the triangle, will pass through all the triangle’s vertices. To draw a triangle’s inscribed circle, construct the triangle’s angle bisectors, which all meet at a point called the incenter. The inscribed circle is centered at the incenter, and its radius is the shortest distance from the incenter to any of the triangle’s sides.

Finally, students have previously measured angles using degrees, but in this unit they learn a new way to measure angles. The radian measure of an angle whose vertex is at the center of a circle is the ratio of the length of the arc defined by the angle to the radius of the circle. That is, \(\theta = \frac{\text{arc length}}{\text{radius}}\). Radian measure will be useful when students study trigonometry in future courses.

Here is a task to try with your student:

A farmer has a circular field, created by a watering system that rotates around a center pivot point. The field’s radius measures 400 meters. As shown in the image, part of the field is planted with oats and part is planted with wheat.

1. Find the area of the field that is planted with oats.
2. A road runs around the circumference of the circle. Find the length of the arc of the road defined by the wheat part of the field.

Solution:

1. The total area of the field is 160,000\(\pi\) square meters because \(\pi(400)^2 = 160,\!000\pi\). The 135-degree sector represents \(\frac38\) of the field because \(\frac{135}{360}=\frac38\). Multiply 160,000\(\pi\) by \(\frac38\) to find an area of 60,000\(\pi\) square meters of oats.
2. The total circumference of the field is \(800\pi\) meters because \(2\boldcdot \pi \boldcdot 400 = 800\pi\). The wheat sector takes up \(\frac58\) of the field because \(1-\frac38=\frac58\). Multiply \(800\pi\) by \(\frac58\) to find that the portion of the road on the outside of the wheat is \(500\pi\), or about 1,571, meters long.