Curriculum Geometry B/ Algebra II
UNIT 4: CIRCLES AND VOLUME
UNDERSTAND AND APPLY THEOREMS ABOUT CIRCLES |
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MGSE9-12.G.C.1 Understand that all circles are similar.
MGSE9-12.G.C.2 Identify and describe relationships among inscribed angles, radii, chords, tangents, and secants. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
MGSE9-12.G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
MGSE9-12.G.C.4 Construct a tangent line from a point outside a given circle to the circle.
MGSE9-12.G.C.2 Identify and describe relationships among inscribed angles, radii, chords, tangents, and secants. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
MGSE9-12.G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
MGSE9-12.G.C.4 Construct a tangent line from a point outside a given circle to the circle.
KEY IDEAS
- A circle is the set of points in a plane equidistant from a given point, which is the center of the circle. All circles are similar.
- A radius is a line segment from the center of a circle to any point on the circle. The word radius is also used to describe the length, r, of the segment.
- A chord is a line segment whose endpoints are on a circle.
- A diameter is a chord that passes through the center of a circle. The word diameter is also used to describe the length, d, of the segment.
- A secant line is a line that is in the plane of a circle and intersects the circle at exactly two points. Every chord lies on a secant line.
- A tangent line is a line that is in the plane of a circle and intersects the circle at only one point, the point of tangency.
- If a line is tangent to a circle, the line is perpendicular to the radius drawn to the point of tangency.
- Tangent segments drawn from the same point are congruent.
- Circumference is the distance around a circle. The formula for circumference C of a circle is C = ∏d, where d is the diameter of the circle. The formula is also written as C = 2∏r, where r is the length of the radius of the circle. ∏ is the ratio of circumference to diameter of any circle.
- An arc is a part of the circumference of a circle. A minor arc has a measure less than 180°. Minor arcs are written using two points on a circle. A semicircle is an arc that measures exactly 180°. Semicircles are written using three points on a circle. This is done to show which half of the circle is being described. A major arc has a measure greater than 180°. Major arcs are written with three points to distinguish them from the corresponding minor arc.
- A central angle is an angle whose vertex is at the center of a circle and whose sides are radii of the circle. The measure of a central angle of a circle is equal to the measure of the intercepted arc.
- An inscribed angle is an angle whose vertex is on a circle and whose sides are chords of the circle. The measure of an angle inscribed in a circle is half the measure of the intercepted arc.
- A circumscribed angle is an angle formed by two rays that are each tangent to a circle. These rays are perpendicular to radii of the circle.
- When an inscribed angle intercepts a semicircle, the inscribed angle has a measure of 90°.
- The measure of an angle formed by a tangent and a chord with its vertex on the circle is half the measure of the intercepted arc.
- When two chords intersect inside a circle, two pairs of vertical angles are formed. The measure of any one of the angles is half the sum of the measures of the arcs intercepted by the pair of vertical angles.
- When two chords intersect inside a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.
- Angles outside a circle can be formed by the intersection of two tangents (circumscribed angle), two secants, or a secant and a tangent. For all three situations, the measure of the angle is half the difference of the measure of the larger intercepted arc and the measure of the smaller intercepted arc.
- When two secant segments intersect outside a circle, part of each secant segment is a segment formed outside the circle. The product of the length of one secant segment and the length of the segment formed outside the circle is equal to the product of the length of the other secant segment and the length of the segment formed outside the circle.
- When a secant segment and a tangent segment intersect outside a circle, the product of the length of the secant segment and the length of the segment formed outside the circle is equal to the square of the length of the tangent segment.
- An inscribed polygon is a polygon whose vertices all lie on a circle.
- For an inscribed quadrilateral, the opposite angles are supplementary.
- When a triangle is inscribed in a circle, the center of the circle is the circumcenter of the triangle. The circumcenter is equidistant from the vertices of the triangle.
- An inscribed circle is a circle enclosed in a polygon, where every side of the polygon is tangent to the circle. Specifically, when a circle is inscribed in a triangle, the center of the circle is the incenter of the triangle. The incenter is equidistant from the sides of the triangle.
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FIND ARC LENGTHS AND AREAS OF SECTORS OF CIRCLES |
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MGSE9-12.G.C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
KEY IDEAS
- Circumference is the distance around a circle. The formula for the circumference, C, of a circle is C=2πr, where r is the length of the radius of the circle.
- Area is a measure of the amount of space a circle covers. The formula for the area, A, of a circle is A = πr², where r is the length of the radius of the circle.
- Arc length is a portion of the circumference of a circle. To find the length of an arc, divide the number of degrees in the central angle of the arc by 360, and then multiply that amount by the circumference of the circle. The formula for the arc length, s, is s=2πr(θ/360), where θ is the degree measure of the central angle and r is the radius of the circle.
IMPORTANT TIP
Do not confuse arc length with the measure of the arc in degrees. Arc length depends on the size of the circle because it is part of the circumference of the circle. The measure of the arc is independent of (does not depend on) the size of the circle. - A sector of a circle is the region bounded by two radii of a circle and the resulting arc between them. To find the area of a sector, divide the number of degrees in the central angle of the arc by 360, and then multiply that amount by the area of the circle. The formula for area of sector is A=πr²(θ/360), where θ is the degree measure of the central angle and r is the radius of the circle.
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EXPLAIN VOLUME FORMULAS AND USE THEM TO SOLVE PROBLEMS |
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MGSE9-12.G.GMD.1 Give informal arguments for geometric formulas.
MGSE9-12.G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
- Give informal arguments for the formulas of the circumference of a circle and area of a circle using dissection arguments and informal limit arguments.
- Give informal arguments for the formula of the volume of a cylinder, pyramid, and cone using Cavalieri’s principle.
MGSE9-12.G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
KEY IDEAS
- The volume of a figure is a measure of how much space it takes up. Volume is a measure of capacity.
- The formula for the volume of a cylinder is V = πr²h, where r is the radius and h is the height. The volume formula can also be given as V = Bh, where B is the area of the base. In a cylinder, the base is a circle and the area of a circle is given by A = πr². Therefore, V = Bh = πr²h.
- When a cylinder and a cone have congruent bases and equal heights, the volume of exactly three cones will fit into the cylinder. So, for a cone and cylinder that have the same radius r and height h, the volume of the cone is one-third of the volume of the cylinder. The formula for the volume of a cone is V = πr²h/3, where r is the radius and h is the height.
- The formula for the volume of a pyramid is V = Bh/3, where B is the area of the base and h is the height.
- The formula for the volume of a sphere is V = 4πr³/3, where r is the radius.
- Cavalieri’s principle states that if two solids are between parallel planes and all cross sections at equal distances from their bases have equal areas, the solids have equal volumes.
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