Second Semester Geometry
Survival Guide
Unit 9: Circles
Big Ideas
9.1
-Radius
-Chord
-Diameter
-Circumference
-Area
-Concentric Circles
-Proving Similarity
-Circle
- Defined as the set of points in a plane that are equidistant from a fixed point called the center
- Is named by the center
- Has three regions: Interior/Exterior/On
-Radius
- A segment whose endpoints are the center of a circle and a point on the circle
- All radii on the same circle are congruent
- labeled as r
-Chord
- A segment whose endpoints are two points on a circle
-Diameter
- A chord that passes through the center
- 2r
-Circumference
- Distance around a circle
- 2πr or πd
-Area
- πr²
-Concentric Circles
- Two circles that share a common center
-Proving Similarity
- Only a translation and dilation are necessary
-Tangent
- A line in the plane of a circle that intersects the circle in exactly one point
- A tangent line is // to the r
- Tangents from a common external point are congruent
-Arc
- Part of a circle
- Congruent arcs: Two equal arcs of the same circle or congruent circles are congruent.
- Minor arc:<180°
- Major Arc:>180°
- Semicircle:=180°
9.3
-Inscribed Angle
- An angle whose vertex is on a circle and whose sides contain chords of the circle
- The measure of an inscribed angle is half the intercepted arc: angle=½arc
- If two inscribed angles intercept the same arc, then the angles are congruent
- A polygon is and inscribed polygon if all of its vertices lie on a circle, the circle that contains the vertices is a circumscribed circle
- If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle
- A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary
9.4
-Tangent Angle
- If a tangent and a cord intersect at a point on a circle, then the measure of each angle formed is half the measure of its intercepted arc
- Angle=½arc
- If two cords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle
- Angle=½(arc1+arc2)
-Exterior Tangent/Secant Angle
- If 2 tangents, 2 secants, or a tangent and a secant intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs
- Angle=½(arc1-arc2)
9.5
-Arc Length Corollary
- In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360°
- Arc Length=arc/360°(2πr)
9.6
-Areas of Sectors
- region bounded by two radii of the circle and their intercepted arc
- The ratio of the area of a sector of a circle is equal to the ratio of the measure of the intercepted arc to 360°
- Sector Area=arc/360°(πr²)
Tips to students
- Remember how ratios work
- Remember formulas for circumference and area
- Remember how to find angle measures with the different formulas
Things I Struggled With
- Remembering the formulas for finding the angle
- I kept practicing it till I could remember.