# Second Semester Geometry

## Big Ideas

9.1

-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

• 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

9.2

-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.
-Types:
• 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
-Interior Angle
• 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.