8 th Grade Geometry Curriculum Map Overview 2016-2017 Unit Number of Days Dates 1 Angles, Lines and Shapes 14 8/2 8/19 2 - Reasoning and Proof with Lines and Angles 14 8/22 9/9 3 - Congruence Transformations and Triangles 18 9/12 10/6 4 - Relationships in Triangles 15 10/17 11/4 5 - Quadrilaterals 26 11/7 12/15 6 - Transformations and Similarity 14 1/4 1/24 7 - Right Triangles and Trigonometry 20 1/25 2/22 8 - Area and Volume with Modeling 19 2/23 3/24 TN Ready Part 1 Testing Window 9 - Circles 15 4/3 4/21 TN Ready Part 2 Testing Window
Unit 1 Angles, Lines and Shapes Standards G.CO.A.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Make Geometric Constructions G.CO.A.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Expressing Geometric Properties with Equations G.GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Essential Understandings 1. A point is a location that has neither shape nor size. 2. A line is made up of points and has no thickness or width. There is exactly one line through two any points. 3. An angle is formed by two noncollinear rays with a common endpoint. 4. A circle is the locus or the set of all points in a plane equidistant from a given point called the center of the circle. 5. Perpendicular lines form rights angles 6. Parallel lines are coplanar lines that do not intersect 7. A line segment is a measurable part of a line that consists of two pints, called endpoints, and all of the points between them. 8. The distance formula is derived from the Pythagorean Theorem and is and can be used to compute perimeters and areas of triangles and rectangles using coordinates. 9. The midpoint formula is used to find the point on a line a segment that divides the segment into two congruent segments and is
Unit 2 Reasoning and Proof with Lines and Angles Standards Prove Geometric Theorems G.CO.C.1 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints Make Geometric Constructions G.CO.A.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Expressing Geometric Properties with Equations G.GPE.B.5 Prove the slope criteria for parallel and perpendicular lines Essential Understandings 1. Complementary angles have a sum of 90. 2. Supplementary angles have a sum of 180. 3. If the noncommon sides of two adjacent angles form a right angle, then they are complementary angles. 4. If two angles form a linear pair, then they are supplementary angles. 5. Vertical angles are two nonadjacent angles formed by two intersecting lines and are congruent. 6. When a transversal intersects two parallel lines, pairs of congruent angles are formed. These include alternate interior angles, corresponding angles, alternate exterior angles 7. The perpendicular bisector in a triangle, a line, segment or ray is the segment that passes through the midpoint of that side and is perpendicular to that side. 8. Slopes of parallel lines are congruent. 9. Slopes of perpendicular lines are opposite reciprocals of each other. 10. The equation of a line is y = mx + b where m is the slope and b is the y-intercept. and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
Unit 3 - Congruence Transformations and Triangles Standards Prove Geometric Theorems G.CO.C.9 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Experiment with Transformations in the Plane G.CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G.CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G.CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand Congruence in Terms of Rigid Motions G.CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Essential Understandings 1. The sum of the measures of the angles of a triangle is 180. 2. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. 3. The acute angles of a right triangle are complementary. 4. There can be at most one right or one obtuse angle in a triangle. 5. The two congruent sides of an isosceles triangle are called legs and the angle whose sides are the legs is called the vertex angle. 6. The side of the isosceles triangle opposite the vertex angle is called the base. 7. The two angles formed by the base and the congruent sides are called the base angles and they are congruent. 8. In two congruent polygons, all the parts of one polygon are congruent to corresponding parts of the other polygon. These corresponding parts include corresponding angles and corresponding sides. 9. A transformation is an operation that maps an original geometric figure, the preimage, onto a new figure, the image. 10. A congruence transformation, also called a rigid transformation or an isometry, is one in which the position of the image may differ from the preimage, but the two figures remain congruent. 11. A reflection is a transformation over a given line called the line of reflection. Each point of the preimage and image are the same distance from the line of reflection. 12. A translation is a transformation that moves all points of the preimage the same distance and direction. 13. A rotation is transformation around a fixed point called
G.CO.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. G.CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. the center of rotation, through a specific angle, and in a specific direction. Each point of the original figure and its image are the same distance from the center. 14. If three sides of a triangle are congruent to three sides of another triangle then they are congruent by Side-Side- Side postulate (SSS). 15. If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then they are congruent by Side-Angle-Side (SAS). 16. If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then they are congruent by Angle-Side-Angle (ASA). 17. If two angles and the nonincluded side of one triangle are congruent to two angles and the nonincluded side of another triangle, then they are congruent by Angle- Angle-Side (AAS).
Unit 4 Relationships in Triangles Standards Prove Geometric Theorems G.CO.C.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints G.CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Essential Understandings 1. If a point is on the perpendicular bisector of a segment, then it is equidistant to the endpoints of the segment. 2. When three or more lines intersect at a common point, they are called concurrent lines and the point of intersection point is called the point of concurrency. 3. The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle and is equidistant from the vertices of the triangle. 4. If a point is on the bisector of an angle, then it is equidistant to the sides of the angle. 5. The angle bisectors of a triangle intersect at a point called the incenter that is equidistant from the sides of the triangle. 6. A median of a triangle is a segment with endpoints being a vertex and the midpoint of the opposite side. 7. The medians of a triangle intersect at a point called the centroid that is two-thirds of the distance from each vertex to the midpoint of the opposite side. The centroid is always inside the triangle. 8. The altitude of a triangle is a segment from the vertex to the line containing the opposite side and perpendicular to the line containing that side. The altitude can be in the interior, exterior, or on the side of the triangle. 9. The lines containing the altitudes of a triangle are concurrent, meeting at a point called the orthocenter. 10. The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle. 11. If one side of a triangle is longer than another side, then the angle opposite the longer side than the angle opposite the shorter side. 12. The sum of the lengths of any two sides of a triangle must be greater than the lengths of the third side.
13. The Hinge Theorem that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.
Unit 5 Quadrilaterals Standards Prove Geometric Theorems G.CO.C.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints G.CO.C.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Use Coordinates to Prove Simple Geometric Relationships G.GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). Essential Understandings 1. The sum of the interior angles of a polygon is (n-2) * 180 where n is the number of sides of the polygon. 2. A parallelogram is a quadrilateral that has both pairs of opposite sides congruent and parallel, both pairs of opposite angles congruent, and its consecutive angles are supplementary. 3. The diagonals of a parallelogram bisect each other. 4. A rectangle is a parallelogram with four right angles and congruent diagonals. 5. A rhombus is a parallelogram with all four sides congruent 6. A trapezoid is a quadrilateral with at least one pair of parallel sides. 7. The midsegment of a trapezoid is a segment that connects the midpoints of the legs of a trapezoid. The midsegment is parallel to each base and is ½ the sum of the lengths of the bases. 8. A kite is a quadrilateral with exactly two pairs of consecutive congruent sides. A kite is not a parallelogram. 9. A quadrilateral can be proven to be a parallelogram if one of the following conditions are met: both pairs of opposite sides are congruent and parallel, both pairs of opposite angles are congruent, one pair of opposite sides are both congruent and parallel, or if the diagonals bisect each other. 10. A parallelogram can be proven to be a rectangle if its diagonals are congruent or if it has right angles. 11. A parallelogram can be proven to be a rhombus if it has four congruent sides. 12. A parallelogram can be proven to be a square if it is both a rhombus and a rectangle.
Unit 6 Transformations and Similarity Standards Use Coordinates to Prove Simple Geometric Theorems Algebraically G.GPE.B6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Understand Similarity in Terms of Similarity Transformations G.SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G.SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar. Explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G.SRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Essential Understandings 1. A ratio is a comparison of two quantities using division. 2. Similar polygons have the same shape but not necessarily the same size. 3. The ratio of the lengths of the corresponding sides of similar polygons is called the scale factor. 4. The corresponding angles of similar polygons are congruent. 5. If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar by Angle-Angle (AA). 6. If the corresponding side lengths of two triangles are proportional, then the triangles are similar by Side- Side-Side Similarity (SSS ) 7. If the lengths of two sides of one triangle are proportional to the lengths of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar by Side- Angle-Side Similarity (SAS ) 8. A dilation is a similarity transformation that enlarges or reduces a figure proportionally with respect to a center point and a scale factor. If the scale factor is greater than one then it is called an enlargement. A scale factor between 0 and 1 is called a reduction. 9. If a line is parallel to one side of a triangle and intersects the other two sides, then it divides the
Prove Theorems Involving Similarity G.SRT.B.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove Geometric Theorems sides into segments of proportional lengths. 10. If two triangles are similar, the lengths of corresponding altitudes are proportional to the lengths of corresponding sides. 11. If two triangles are similar, the lengths of corresponding angle bisectors are proportional to the lengths of corresponding sides. 12. If two triangles are similar, the lengths of corresponding medians are proportional to the lengths of corresponding sides. G.CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
Unit 7 Right Triangles and Trigonometry Standards Prove Geometric Theorems G.CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define Trigonometric Ratios and Solve Problems Involving Right Triangles G.SRT.C.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G.SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles. G.SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply Trigonometry to General Triangles G.SRT.D.9 (+) Derive the formula A = 1/2 ab sin(c) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. G.SRT.D.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems. G.SRT.D.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). Essential Understandings 1. The geometric mean of two numbers is the positive square root of their product. 2. If an altitude is drawn to the hypotenuse of a right triangle, then the triangles formed are similar to the original triangle and each other. 3. The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of the altitude is the geometric mean between the lengths of these two segments. 4. The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of a leg of the triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg. 5. In a right triangle, the sum of the squares of the legs is equal to the square of the length of the hypotenuse. This relationship can be represented by a 2 + b 2 = c 2 where a and b are the legs and c is the hypotenuse and is known as the Pythagorean Theorem. 6. A Pythagorean triple is a set of three non-zero whole numbers that satisfy the Pythagorean Theorem. (3, 4, 5; 5, 12, 13; 8, 15, 17, etc.) 7. In a 45-45 -90 triangle, the legs are congruent and the hypotenuse is 2 times the length of a leg. 8. In a 30-60 -90 degree triangle, the length of the hypotenuse is 2 times the length of the shorter leg and the longer leg is 3 times the length of the shorter leg. 9. A trigonometric ratio is the ratio of two sides of a right triangle. 10. The sine of an acute angle in a right triangle is the ratio of the length of the leg opposite the angle and the length of the hypotenuse. 11. The cosine of an acute angle in a right triangle is the ratio
of the length of the leg adjacent to the angle and the length of the hypotenuse. 12. The tangent of an acute angle in a right triangle is the ratio of the length of the leg opposite the angle and the leg adjacent to the angle. 13. The sine of one acute angle in a right triangle is the same as the cosine of its complement. 14. The area of a triangle can be found by using the formula A = 1/2 ab sin(c). 15. The Law of Sines and Law of Cosines can be used to find unknown measurements in both right and non-right triangles. 16. The Law of Sines can be represented by 17. The Law of Cosines can be represented by a 2 = b 2 + c 2 2bc cos A b 2 = a 2 + c 2 2ac cos B c 2 = a 2 + b 2 2ab cos C 18. A vector describes both the magnitude and direction of a real number.
Unit 8 Area and Volume with Modeling Standards Explain Area and Volume Formulas and Use Them to Solve Problems G.GMD.A.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri s principle, and informal limit arguments. G.GMD.A.2 (+) Give an informal argument using Cavalieri s principle for the formulas for the volume of a sphere and other solid figures. G.GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Visualize Relationships between Two-Dimensional and Three-dimensional Objects G.GMD.B.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Translate Between the Geometric Description and the Equation for a Conic Section Essential Understandings 1. The area of a circle is equal to times the square of the radius (r) or * r 2 2. The volume of a cylinder is r 2 h where r is the radius and h is the height of the cylinder 3. The volume of a pyramid is 1/3 B*h where B is the area of the base and h is the height of the pyramid. 4. The volume of a cone is 1/3 B*h or 1/3 r 2 h where B is the area of the base, h is the height of the cone and r is the radius of the base. 5. The volume of a sphere is 4/3 r 3 where r is the radius of the sphere. 6. Cavalieri's Principle states that if two solids have equal altitudes, and if sections made by planes parallel to the bases and at equal distances from them are always in a given ratio, then the volumes of the solids are also in this ratio. The formula is B*h where B is the area of a cross section and h is the height of the solid. 7. A cross section is the intersection of a body in threedimensional space with a plane. G.GPE.A.2 Derive the equation of a parabola given a focus and directrix. G.GPE.A.3 (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. 8. The equation of a parabola is ax 2 + bx + c in standard form and a(x-h) 2 + k in vertex form
Apply Geometric Concepts in Modeling Situations 9. The equation of an ellipse is G.MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). G.MG.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). G.MG.A.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). where: x,y are the coordinates of any point on the ellipse, a, b are the radius on the x and y axes respectively 10. The equation of a hyperbola is where x 0 and y 0 are the center points and a and b are the major and minor axes.
Unit 9 Circles Standards Understand and Apply Theorems about circles G.C.A.1 Prove that all circles are similar. G.C.A.2 Identify and describe relationships among inscribed angles, radii, and chords. 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. G.C.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. G.C.A.4 (+) Construct a tangent line from a point outside a given circle to the circle. Explain Area and Volume Formulas and Use Them to Solve Problems G.GMD.A.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri s principle, and informal limit arguments. Make Geometric Constructions G.CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Find Arc Lengths and Areas of Sectors of Circles G.C.B.5 Derive using similarity the fact that the length of the arc Essential Understandings 1. A circle is the locus or the set of all points in a plane equidistant from a given point called the center of the circle. 2. A radius of a circle is a segment with endpoints at the center and on the circle. 3. A chord is a segment with endpoints on the circle. 4. A diameter of a circle is a chord that passes through the center of the circle. 5. If two circle have congruent radii, then they are congruent circles. 6. All circles are similar. 7. Concentric circles are coplanar and have the same center. 8. The circumference of a circle is the distance around the circle. The ratio of the circumference to the diameter is represented by the symbol. 9. A central angle of a circle is an angle with a vertex in the center of the circle and sides as radii. 10. An arc is a portion of a circle defined by two endpoints. 11. A minor arc is the shortest arc connecting two endpoints on the circle. Its measure is less than 180 and equal to the measure of its related central angle. 12. A major arc is the longest arc connecting two endpoints on the circle. Its measure is greater than 180 and equal to 360 minus the measure of the minor arc with the same endpoints. 13. A semicircle has endpoints that lie on the diameter and has a measure of 180. 14. Arc length is the distance between the endpoints along an
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. Translate Between the Geometric Description and the Equation for a Conic Section G.GPE.A.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation arc measured in linear units. 15. The ratio of the length of an arc to the circumference of the circle is equal to the ratio of the degree measure of the arc to 360. 16. A sector of a circle is a region of a circle bounded by a central angle and its intercepted arc. 17. The ratio of the area of a sector to the area of the whole circle is equal to the ratio of the degree measure of the intercepted arc to 360. 18. Radian measure is a way to measure angles. One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle. 19. In the same circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. 20. If a diameter (or radius) of a circle is perpendicular to a chord, then it bisects the chord and its arc. 21. The perpendicular bisector of a chord is a diameter (or radius) of the circle. 22. In the same circle or in congruent circles, two chords are congruent if and only if they are equidistant to the center. 23. An inscribed angle has a vertex on the circle and sides that contain chords of the circle. 24. An inscribed angle on a diameter is a right angle. 25. An intercepted arc has endpoints on the sides of an inscribed angle and lies in the interior of the inscribed angle. 26. If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
27. A tangent is a line in the same plane as a circle that intersects the circle in exactly one point called the point of tangency. 28. In a plane, a radius is perpendicular to the tangent line when drawn to the point of tangency. 29. If two segments from the same exterior point are tangent to a circle, then they are congruent. 30. A secant is a line that intersects a circle in exactly two points. 31. If two secants or chords intersect in the interior of the circle, then the measure of the angle formed is ½ the sum of the measure of the arcs intercepted by the angle and its vertical angle. 32. If a secant and a tangent intersect at the point of tangency, then the measure of angle formed is ½ the measure of the intercepted arc. 33. If two secants, a tangent and a secant, or two tangents intersect in the exterior of the circle, then the measure of the angle formed is ½ the difference of the measures of the intercepted arcs. 34. If two chords intersect in a circle, then the products of the lengths of the chords are equal. 35. If two secants intersect in the exterior of a circle, then the product of the measure of one secant segment and its external secant segment is equal to the product of the measures of the other secant and its external secant segment. 36. If a tangent and a secant intersect in the exterior of a circle, then the square of the measure of the tangent is equal to the product of the measures of the secant and its external segment. 37. The equation of a circle is derived from the Pythagorean Theorem and is represented by:
(x-h) 2 + (y-k) 2 = r 2 where r is the radius of the circle, and h,k are the coordinates of its center