Section 10.1 - Congruence Through Constructions Definitions: Similar ( ) objects have the same shape but not necessarily the same size. Congruent ( =) objects have the same size as well as the same shape. circle is the set of all points in a plane equidistant from a given point, its center. The distance is the radius. The word radius is used to describe both the segment from the center to a point on the circle and the length of that segment. n arc of a circle is any part of the circle that can be drawn without lifting a pencil. The center of an arc is the center of the circle containing the arc. Given any two points on a circle, the short arc is known as the minor arc and the longer arc is known as the major arc. If the major arc and the minor arc are the same size, each is a semicircle. If only two points are used in naming an arc, the minor arc is implied. Circle Construction: Given a center, O, and a radius, PQ 1. Set the legs of the compass on P and Q to measure PQ. 2. Keeping the distance determined, set the compass pointer at the center, O, and move the pencil to draw the circle. Example 1: Construct a circle with center, O, and radius, PQ P Q O 1
Segment Construction: Given a segment, PQ 1. Set the legs of the compass on P and Q to measure PQ. 2. Keeping the distance determined, place the point of the compass at any point C on line l and strike an arc to locate point D. CD = PQ Example 2: Construct a line segment congruent to PQ. P Q l Triangle Congruence: BC is congruent to DEF, written BC = DEF, if and only if = D, B = E, C = F, B = DE, BC = EF, and C = DF. B E C D Corresponding Parts of Congruent Triangles are Congruent (CPCTC) F Side, Side Side Property (SSS): If three sides of one triangle are congruent, respectively, to three sides of a second triangle, then the triangles are congruent. Constructing a Triangle Given Three Sides: To construct a B C = BC using the three sides: Draw a ray X Construct a segment C = C. Construct a circle (or arc) with center at and radius B and a circle (or arc) with center at C and radius BC. Choose a point of intersection of the two circles (arcs) and label it B Draw B and B C. By construction and SSS, BC = B C. Example 3: Construct B C = BC. B C 2
Triangle Inequality: The sum of the measures of any two sides of a triangle must be greater than the measure of the third side. Constructing Congruent ngles: To copy B so that one of its sides is B X: With center at B, mark off an arc C to form isosceles triangle BC. Make an arc with same radius with center at B. Label the intersection of B X and arc C as C. With pointer at C, mark an arc C so that C = C. Draw B. B = B because BC = B C. Example 4: Copy B so that one of its sides is B X. B B X Question: How many different triangles can be constructed from two given segments? Side, ngle, Side Property (SS): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, respectively, then the two triangles are congruent. Constructions Involving Two Sides and an Included ngle of a Triangle: To construct B C given B, C and Draw a ray X. Copy C onto ray Xto get C. Copy at on C to get ray Y. Copy B onto ray Y to get B. Draw B C. Example 5: Construct B C = BC given B, C, and : C B 3
Example 6: If, in two triangles, two sides and an angle not included between these sides are congruent, respectively, determine whether the triangles must be congruent. Definitions: The perpendicular bisector of a segment is a segment, ray, or line that is perpendicular to the segment at its midpoint. n altitude of a triangle is the perpendicular segment from a vertex of the triangle to the line containing the opposite side of the triangle. Example 7: Let s explore an isosceles triangle. Theorem 10-1: The following holds for every isosceles triangle: The angles opposite the congruent sides are congruent. (Base angles of an isosceles triangle are congruent.) The angle bisector af an angle fromed by two congruent sides contains an altitude of the triangle and is the perpendicular bisector of the third side of the triangle. Theorem 10-2: ny point equidistant from the endpoints of a segment is on the perpendicular bisector of the segment. ny point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. 4
Construction of the Perpendicular Bisector of a Segment: To construct a perpendicular bisector of B: Put the compass point on and the pencil point anywhere past the midpoint of B. Draw a circle (or arcs above and below) with as its center. Keeping the distance determined, draw a circle (or arcs above and below) with B as its center. Draw a segment, ray, or line through the two intersection points. Example 8: Construct the perpendicular bisector of the segment B B Construction of a Circle Circumscribed bout a Triangle: circle is circumscribed about a triangle when all three vertices of the triangle are on the circle. The circle is called a circumcircle. Its center is called the circumcenter and its radius is called the circumradius To construct a circle circumscribed about BC: Construct the perpendicular bisectors of B and C. Construct a circle with radius D and center D where D is the intersection of the perpendicular bisectors. Example 9: Construct a circle circumscribed about BC. B C Section 10.1 Homework Problems: 1, 3-5, 7, 9, 17-21, 24, 27-29, 35, 37, 43 5
Section 10.2 - Other Congruence Properties ngle, Side, ngle Property (S): If two angle and the included side of one triangle are congruent to two angles and the included side of another triangle, respectivley, then the triangles are congruent. ngle, ngle, Side Property (S): If two angles and a corresponding side of one triangle are congruent to two angles and a corresponding side of another triangle, respectively, then the two triangles are congruent. Example 1: Construct a triangle given the two angles and included side below: B B Example 2: Prove that the opposite sides of a parallelogram are congruent. 6
Example 3: Prove that the opposite angles of a parallelogram are congruent. Example 4: Prove that the diagonals of a parallelogram bisect each other. 7
Example 5: Prove that the consecutive angles between parallel sides of a trapezoid are supplementary. Section 10.2 Homework Problems: 2, 3, 5, 6, 9, 12, 16, 26, 31, 32, 39-42 8
Section 10.3 - Other Constructions We use the definition of a rhombus and the following properties (from the table in Section 10.2) to accomplish basic compass-and-straightedge constructions: 1. rhombus is a parallelogram in which all the sides are congruent. 2. quadrilateral in which all the sides are congruent is a rhombus. 3. Each diagonal of a rhombus bisects the opposite angles. 4. The diagonals of a rhombus are perpendicular. 5. The diagonals of a rhombus bisect each other. Constructing Parallel Lines Given a line l and a point P not on l, construct a line through P parallel to l. Rhombus Method: 1. Through P, draw any line that intersects l. Label this intersection. P will be a side of the rhombus. 2. Draw an arc with the pointer at and with radius P to mark the third vertex, X, of the desired rhombus. 3. With the same opening of the compass, draw intersecting arcs, first with the pointer at P and then with the pointer at X to find Y, the fourth vertex of the rhombus. 4. Draw PY. We see that PY is parallel to l as requested. Example 1: Construct a line through P parallel to l using the Rhombus Method. P l Corresponding ngle Method: 1. Through P, draw a line that intersects l. 2. Label the angle formed α. 3. Copy angle α at point P. We see that PQ is parallel to l as requested. Example 2: Construct a line through P parallel to l using the Corresponding ngle Method. P l 9
Constructing ngle Bisectors n angle bisector is a ray that separates an angle into two congruent angles. To construct the angle bisector of : 1. With the pointer at, draw any arc intersecting the angle. Label the intersection points B and C. We now have three vertices of the rhombus:, B, and C. 2. Draw an arc with center at B and radius B. 3. Draw an arc with center at C and radius B. 4. Label the intersection of the two arcs D, the fourth vertex of the rhombus. 5. Draw D, the angle bisector of. Example 3: Construct the angle bisector of. Example 4: Construct the angle bisector of. 10
Constructing Perpendicular Lines Constructing a perpendicular to a line from a point not on the line: 1. Draw an arc with center at P that intersects the line at two points. Label these points and B. These points,, B, and P, are three of the vertices of our rhombus. 2. With the same compass opening, make two intersecting arcs, one with center at and the other with center at B. Label the intersection of these arcs Q, the final vertex of our rhombus. 3. Draw PQ, our requested line perpendicular to l. Example 5: Construct a line perpendicular to l through P. P l Constructing a perpendicular to a line from a point on the line 1. Draw any arc with center at M that intersects l in two points. Label the points of intersection and B. B will be the diagonal of a rhombus. 2. Use a larger opening for the compass and draw intersecting arcs, with centers at and B. Label the points of intersection C and D. This is the other diagonal of our rhombus. 3. Draw CD, the requested line. Example 6: Construct a line perpendicular to l through M. M l 11
Example 7: Construct an altitude from vertex in the triangle below. B C Properties of ngle Bisectors Theorem 10-3: a) ny point P on an angle bisector is equidistant from the sides of the angle. b) ny point that is equidistant from the sides of an angle is on the angle bisector of the angle. Note: The distance from a point to a line is the length of the perpendicular segment from the point to the line. Example 8: Prove Theorem 10-3(a). 12
Constructing a Circle Inscribed in a Triangle: Note: line is tangent to a circle if it intersects the circle in one and only one point and is perpendicular to a radius. circle is inscribed in a triangle if all the sides of the triangle are tangent to the circle. The inscribed circle is the incircle; the center is the incenter. Bisect two angles of the triangle. The intersection of the angle bisectors, P, will be the center of the circle. Construct a perpendicular from P to a side of the triangle. The length of that segment will be the length of the radius of the circle. Example 9: Inscribe a circle in BC. B C Section 10.3 Homework Problems: 3-5, 8, 10, 21, 23-25, 41, 42 13
Section 10.4 - Similar Triangles and Similar Figures Two figures that have the same shape but not necessarily the same size are similar ( ). Example 1: Use your protractor to measure each side and angle in the triangles below and summarize your findings. B E C D F Definition of Similar Triangles: BC is similar to DEF, written BC DEF, if and only if = D, B = E, C = F, and B DE = C DF = BC EF. Note: We say that the corresponding sides are proportional and refer to the ratio of the corresponding side lengths as the scale factor. Tests for Similar Triangles: We can conclude that BC PQR if at least one of the following conditions is true: 1. = P and B = Q ( test) 2. 3. PQ B = QR BC = RP (Three pairs of corresponding sides are proportional) C PQ B = RP C and = P (Two pairs of sides are proportional and included angles are congruent) 14
Example 2: Which of the following pairs of triangles are similar? Example 3: In the figure below, given BD = 12, solve for x. Properties of Proportion: Theorem 10-4: If a line parallel to one side of a triangle intersects the other sides, then it divides those sides into proportional segments. Example 4: Prove Theorem 10-4. 15
Theorem 10-5: If a line divides two sides of a triangle into proportional segments, then the line is parallel to the third side. Example 5: Prove Theorem 10-5. Theorem 10-6: If parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on any transversal. Example 6: Prove Theorem 10-6. Construction Separating a Segment into Congruent Parts To separate B into a given number of congruent parts: 1. Draw any ray, C, such that, B, and C are noncollinear. 2. Mark off the given number of congruent segments (of any size) on C. In this case, we use three congruent segments. Label the points 1, 2, and 3. 3. Connect B to 3. 4. Through 2 and 1, construct parallels to B 3. Example 7: Separate B into three congruent segments. 16
Midsegments of Triangles and Quadrilaterals: midsegment is a segment that connects midpoints of two adjacent sides of a triangle or quadrilateral. Theorem 10-7: The Midsegment Theorem: The midsegment (segment connecting the midpoints of two sides of a triangle) is parallel to the third side of the triangle and half as long. Example 8: Prove Theorem 10-7. Theorem 10-8: If a line bisects one side of a triangle and is parallel to a second side, then it bisects the third side and therefore is a midsegment. Example 9: Prove Theorem 10-8. 17
Indirect Measurements Similar triangles have long been used to make indirect measurements by using ratios involving shadows. Example 10: On a sunny day, a tall tree casts a 40 meter shadow. t the same time, a meter stick held vertically casts a 2.5 meter shadow. How tall is the tree? Section 10.4 Homework Problems: 1, 2, 4, 5, 8-10, 12, 24, 25, 36, 42 18
Section 10.5 - Trigonometry Ratios via Similarity We come to the conclusion that if the angle changes, the ratio changes too, or in other words, this ratio is the function of the angle of elevation. Since any two right triangles with the same acute angle are similar, the ratio of two sides of any of these similar triangles does not depend on the size of the triangle, it only depends on the angle. Trigonometric Ratios In any right triangle BC with a right angle at C, we refer to b = C as the side adjacent to, a = BC as the side opposite, and c = B as the hypotenuse. Each of the ratios side opposite hypotenuse side adjacent hypotenuse side opposite side adjacent does not depend on the size of the triangle, it depends only on the size of BC with C = 90, we define sin( ) = side opposite hypotenuse cos( ) = side adjacent hypotenuse tan( ) = side opposite side adjacent Example 1: For each of the following figures, solve for x. In a right triangle 19
Note: You are responsible for remembering the sine, cosine, and tangent of 3 special angles, 30, 45, and 60. Example 2: Let s look at the sin, cos, and tan of these 4 special angles. Example 3: For each of the following figures, find the exact value of x. 20
Example 4: Show that tan( ) = sin( ) cos( ) Example 5: Use the Pythagorean Theorem a 2 + b 2 = c 2 (here a and b are the legs and c is the hypotenuse of a right triangle) to show that sin 2 +cos 2 = 1. Section 10.5 Homework Problems: 1-4, 9, 14 21
Section 10.6 - Lines in a Cartesian Coordinate System Cartesian Coordinate System is constructed by placing two number lines perpendicular to each other. The intersection point of the two lines is the origin, the horizontal line is the x-axis, and the vertical line is the y-axis. The location of any point can be described by an ordered pair of numbers (a,b). The first component is the x-coordinate; the second component is the y-coordinate. Example 1: Plot the following points on a Cartesian Coordiante System. Equations of Vertical and Horizontal Lines Vertical Line: The graph of the equation x = a, where a is some real number, is a line perpendicular to the x-axis through the point with coordinates (a,0). Horizontal Line: The graph of the equation y = b, where b is some real number, is a line perpendicular to the y-axis through the point with coordinates (0,b). Example 2: Sketch the graph for each of the following equations: Questions: What is the equation of the x-axis? What is the equation of the y-axis? 22
Equations of Lines Every line has an equation of the form either y = mx+b or x = a, where m is the slope and b is the y-intercept. Example 3: Find the equation of the line with slope 2 and y-intercept 4. Slope Formula: Given two points (x 1,y 1 ) and B(x 2,y 2 ) with x 1 x 2, the slope m of the line B is m = y 2 y 1 x 2 x 1 Note: ny two parallel lines have the same slope, or are vertical lines with undefined slope. Example 4: Find the slope of the line passing through the points ( 4,6) and (2, 3). Graph the line. Point-Slope Form of the Equation of the Line: The equation of the line that passes through the point (x 1,y 1 ) and has slope m is given by: y y 1 = m(x x 1 ) Example 5: Find the equation of the line passing through the points ( 4,6) and (2, 3). Example 6: Find the x and y intercepts of the line with equation y = 3x+8. 23
Systems of Linear Equations What are the possible cases when solving a system of linear equations? Example 7: Solve the following system of linear equations: 4x 5y = 30 2x+y = 8 Example 8: Solve the following system of linear equations: 2x y = 1 6x 3y = 12 24
Example 9: Solve the following system of linear equations: 3x 7y = 4 6x 14y = 8 Example 10: Diego s piggy bank contains 55 coins. If all of the coins are either nickels or dimes and the value of the coins is $4.75, how many of each kind of coin are there? Section 10.6 Homework Problems: 1-7, 9, 10, 22, 24, 26, 43 25