G.GMD.3 Some Things To Know

Transcription

1 1 GGMD3 Some Things To Know The volume formula for a right prism or cylinder is, where B is the area of the base and h is the height For a cone or pyramid, the volume formula is 1 You try: A grain silo is in the shape of a cylinder The silo has a diameter of 12 feet and a height of 20 feet Farmer Juan has reaped 10,000 cubic feet of wheat How many silos would he need to fit all his wheat? For a sphere: Example: Abigail is putting juice into cone shaped containers to make popsicles She has approximately one quarter of a gallon, or 58 cubic inches, of juice to make the popsicles with Each container has a 3-inch height and 2-inch radius Find the maximum number of popsicles that Abigail could make with 58 cubic inches of juice Solution: Find the volume for each cone-shaped container: Base is a circle, so: cubic inches Since Abigail has 58 cuin of juice, divide that number by the volume of each container (1256 cuin) to find the number of containers she can fill This means that she could fill 4 containers completely, but she doesn t have enough to fill 5 containers GGMD3 GGMD3 Page 1 of 12 MCC@WCCUSD 01/23/15

2 2 GGPE4 Some Things To Know Use the distance formula and the slope formula to help you determine which geometric shape is created when given the vertices coordinates The distance between two points can be found with the equation 2 You try: Triangle RST is graphed on a coordinate plane with vertices R( 2, 2), S(5, 1), and T( 1, 4) What type of triangle is formed? The slope formula is Example: Quadrilateral PQRS is graphed on a coordinate plane with vertices located at P(1, 5), Q(3, 4), R(0, 2), and S( 2, 1) What type of quadrilateral is formed? Solution: Step 1: Find the length of each side: Doing the same with the other three sides, we find:,, So, both pairs of opposite sides are congruent Step 2: Find the slope of each line: Doing the same for the other three sides, we find:,, Since the adjacent sides slopes are negative reciprocals, the lines are perpendicular, making right angles at each of the four vertices Step 1 and Step 2 conclusions lead to knowing that the vertices make a rectangle GGMD4 GGMD4 Page 2 of 12 MCC@WCCUSD 01/23/15

3 3 GGPE5 Some Things To Know --The slope formula is m = y 2 y 1 x 2 x 1 ---Slopes of parallel lines are equal ---Slopes of perpendicular lines are opposite reciprocals Example: Given the two points L(3, 2) and M( 4,8), which of the following statements are true? Select all that apply A The slope of the line LM is 6 B The slope of the line LM is 10 7 C The line y = 1 x 3 is parallel to LM 6 D Line y = 7 x +8 is perpendicular to LM 10 E The line 10x + 7y = 35 is parallel to LM F Line x 6y = 42 is perpendicular to LM Solution: Find the slope of LM : 3 You try: Given the two points G(6, 4) and H( 1,3) and the line 2x + 2y = 7, which of the following statements are true? Select all that apply A The slope of the line GH is -1 B The slope of the line GH is 1 7 C The line y = 7x + 4 is parallel to GH D The line y = x 8 is parallel to GH E The line x + y = 6 is perpendicular to GH F The line x + 7y =12 is perpendicular to GH m = y y = x 2 x 1 3 ( 4) = So, A is not correct, but B is correct --Parallel lines have equal slopes, so C is not correct, but D is correct since 7 is the negative 10 reciprocal of 10 7 so the lines are perpendicular --For E and F, solve each equation for y to determine the slope of the line: 10x + 7y = 35 x 6y = 42 7y = 35 10x 6y = 42 + x y = x y = x E is correct since parallel lines have equal slopes and F is incorrect since the slope of the perpendicular line must be 7 10 GGPE5 GGPE5 Page 3 of 12 MCC@WCCUSD 01/23/15

4 4 WCCUSD Geometry GGPE6 Some Things To Know Finding a point on a given line segment that is partitioned by a certain ratio requires knowledge of a few things: The partition ratio and the scale factor are directly related (eg a partition ratio of 5:1 has a scale factor of ) The run is the horizontal distance between the segment s endpoints The rise is the vertical distance between the segment s endpoints The partitioning point s coordinate is 4 You try: A directed line segment is graphed below Plot point R on the graph so that it splits WY into segments with a ratio of 1:2 factor Explain how you determined your answer Example: Given the point P that partitions the directed line segment QR into a ratio of 1:4 with Q( 5, 4) and R(0, 6), select all statements below that lead to finding the point P 1 A The scale factor is 5 B The run is 5 C The rise is 10 " D (5), # $ 5 (10) % & ' Solution: All answers are correct A Since the ratio is 1:4, there are 5 total partitions and the 1 tells you are identifying the first point, which helps identify your scale factor of 1 5 B The x-coordinate is increasing by 5, so the run is 5 C The y-coordinate is increasing by 10, so the rise is 10 D The values were correctly substituted GGPE6 GGPE6 Page 4 of 12 MCC@WCCUSD 01/23/15

5 5 GC2 Some Things To Know For circles: A central angle is equal in measure to its intercepted arc An inscribed angle is half the measure of its intercepted arc A line tangent to a point on the circle makes a right angle with a radius/diameter that meets at the point of tangency 5 You try: Given Circle A below with EF tangent to Circle A at point D and tangent to Circle G at point F, if m BCD = 62 and mbc = 83, which of the following statements are true? Example: Given Circle C below with tangent to the circle at point B, if which of the following statements are true? A mab! = 30 B mab! = 60 C D E F F A m DAB = 62 B m DAB =124 C m ADF = m DFG D m EDC = 90 E mcbd = 207 F mcd = 153 G CDA and CDF are complementary Solution: mab! = 60 is correct because its inscribed angle is 30 An intercepted arc is twice the measure of its inscribed angle is correct because that angle intercepts AB!, which is 60 Inscribed angles are half the measure of its intercepted arcs There is not enough information to determine the measure of is correct because it is a central angle and its intercepted arc has the same measure is correct because any tangent line is perpendicular to the radius at the point of tangency GC2 GC2 Page 5 of 12 MCC@WCCUSD 01/23/15

6 6 GC3 Some Things To Know To circumscribe a circle about a triangle: 1 Construct the perpendicular bisectors of each side Draw them long enough to intersect each other (the dashed lines in the figure below) 2 The intersection point of the bisectors is the center of your circle Open your compass with its point at the circle s center and draw the circle so that the triangle s vertices all connect 6 You try: Construct a circle that circumscribes the triangle below A B C B A C To inscribe a circle within a triangle: 1 Construct the angle bisectors of (at least) two angles The intersection point will be the center of your circle 2 From this point, construct a line perpendicular to one of the sides of the triangle This will determine the point of tangency Draw the circle Construct a circle that inscribes the triangle below B B A A C C GC3 GC3 Page 6 of 12 MCC@WCCUSD 01/23/15

7 7 GGPE1 Some Things To Know The equation of a circle is: (x h)! + (y k)! = r! where r represents the distance of all points from the center of the circle, (h, k) Given the equation of a circle in the standard form, complete the square for both x- and y-terms to find the center of the circle Example: Draw the circle whose equation is (x 4)! + (y + 2)! = 16 Solution: Step 1: Determine the center: h is 4 and k is -2 (substituting 2 into y k would yield y + 2) Step 2: Determine the radius Since r! = 16, the radius must be 4 Step 3: Plot your center on the graph and map out four points that are 4 units away to help draw your circle 7 You Try: Draw the circle whose equation is (x + 1)! + (y 2)! = 25 Explain how you determined your answer You Try: Complete the square to find the center and radius of the circle whose equation is x 2 + y 2 4x +8y +11= 0 Example: Complete the square to find the center and radius of the circle whose equation is x 2 + y 2 + 2x 6y +1= 0 Solution: x 2 + y 2 + 2x 6y +1= 0 x 2 + y 2 + 2x 6y = 1 x 2 + 2x + y 2 6y = 1 (x 2 + 2x + )+ (y 2 6y + ) = 1 (x 2 + 2x + ( 1) 2 )+ (y 2 6y + (3) 2 ) = 1+ ( 1) 2 + (3) 2 (x +1) 2 + (y 3) 2 = 9 Move constant to right side Commute x- and y-terms together Complete the square Factor The center is at ( 1,3) and the radius is 3 GGPE1 GGPE1 Page 7 of 12 MCC@WCCUSD 01/23/15

8 8 GGPE4 Some Things To Know The Distance Formula ( ) can be used to determine whether a given point is inside, on or outside a given circle Use the circle s center coordinates and the given point in question and substitute into the Distance Formula 8 You Try: Is the point (5, 8)inside, on or outside the circle (x 3)! + (y + 5)! = 81? Use the Midpoint Formula to find the center of a circle given the endpoints of the diameter Example: Is the point ( 3, 4) inside, on or outside the circle (x 6)! + (y + 1)! = 100? Solution: The center of the circle is (6, 1) Substituting this point and the point in question into the Distance Formula: d = ( 3 6) 2 + (4 ( 1)) 2 d = ( 9) 2 + (5) 2 d = d = This is a little more than the radius of 10 units, so the point is outside the circle You Try: Given the endpoints of the diameter of a circle, (10, 5) and ( 2, 9)find the equation of the circle Example: Given the endpoints of the diameter of a circle, ( 4, 2) and (4,8) find the equation of the circle Solution: Find the center of the circle using the midpoint formula: " $ # 4 + 4, Find the radius: d = (4 0) 2 + (8 3) 2 d = (4) 2 + (5) 2 % & '! 0 2, 6 $ # " 2 & 0,3 % ( ) Center d = d = 41 Radius GGPE4 The equation is x 2 + (y 3) 2 = 41 End of Study Guide GGPE4 Page 8 of 12 MCC@WCCUSD 01/23/15

9 1 You Try Solutions You try: A grain silo is in the shape of a cylinder The silo has a diameter of 12 feet and a height of 20 feet Farmer Juan has reaped 10,000 cubic feet of wheat How many silos would he need to fit all his wheat? If d = 12, then r = 6 2 You try: Triangle RST is graphed on a coordinate plane with vertices R( 2, 2), S(5, 1), and T( 1, 4) What type of triangle is formed? Find the side lengths: h = 20 Base is a circle, so cuft (each silo holds this amount) He reaped 10,000 cu ft of wheat So, dividing this by the amount each silo holds results in: No sides have the same measure, so the triangle is scalene Now check the slopes of each side: m RS = 2 ( 1) 2 5 = 1 7 = , m ST = 1 ( 4) 5 ( 1) = 3 6 = 1 2 which means he would need 5 silos to hold his wheat m RT = 2 ( 4) 2 ( 1) = 2 1 = 2 GMD3 Since the slopes 1 2 and 2 are opposite reciprocals, those two lines are perpendicular, forming a right angle So the triangle formed is a right scalene triangle GGMD4 Page 9 of 12 MCC@WCCUSD 01/23/15

10 3 You try: Given the two points G(6, 4) and H( 1,3), which of the following statements are true? Select all that apply A The slope of the line GH is -1 B The slope of the line GH is 1 7 C The line y = 7x + 4 is parallel to GH D The line y = x 8 is parallel to GH E The line x + y = 6 is perpendicular to GH F The line x + 7y =12 is perpendicular to GH 4 You try: Directed line segment is graphed below Plot point R on the graph so that it splits WY into segments with a ratio of 1:2 R The slope of the line containing the two points is: m = ( 1) = 7 7 = 1 So A is correct D is also correct since the slope of that line is also 1 Solving x + y = 6 for y, we find y = x + 6, which has a slope of 1 This is the opposite reciprocal of -1, so the line is perpendicular to GH So, E is correct Explain how you determined your answer The starting point is W ( 2, 4) First find the rise and run The rise is 9 and the run is 6 The ratio segmentation is 1:2, which gives a scale factor of 1 3 Now, use ( a + k(run),b + k(rise) ) to find the point in question: " $ (6), # 3 (9) % ' & GGPE5 ( 2 + 2, 4 + 3) This is the point that will partition the given line segment into a ratio of 1:2 GGPE6 Page 10 of 12 MCC@WCCUSD 01/23/15

11 5 You try: Given Circle A below with EF tangent to Circle A at point D and tangent to Circle G at point F, if m BCD = 62 and mbc = 83, which of the following statements are true? A m DAB = 62 B m DAB =124 C m ADF = m DFG D m EDC = 90 E mcbd = 207 F mcd = 153 G CDA and CDF are complementary If m BCD = 62, then mbd = 124 since an intercepted arc is twice its inscribed angle This means that m DAB =124 since a central angle has the same measure as its intercepted arc So B is correct Both ADF and DFG are right angles since they are created by a tangent line and a radius of each circle, which form perpendicular lines So C is correct D is not correct because DC is not part of a radius If mbd = 124 and mbc = 83 then mcbd = 207 since this major arc is the sum of the two minor arcs E is correct mcd = 153 because this is what remains of a complete circle of 360 F is correct CDA and CDF are complementary because they form a right angle G is correct 6 You try: Construct a circle that circumscribes the triangle below Construct the perpendicular bisectors of at least two of the sides Their intersection is the center of the circle (circumcenter) Open your compass such that the point is on this center and the pencil point is on a vertex of the triangle Draw your circle Construct a circle that inscribes the triangle below Bisect two of the angles Then use the intersection point (which is the circle s center) to construct a line perpendicular to a side This will determine one of the points of tangency Use your compass to draw your circle GC2 GC3 Page 11 of 12 MCC@WCCUSD 01/23/15

12 7 You Try: Draw the circle whose equation is (x + 1)! + (y 2)! = 25 8 You Try: Is the point (9, 12)inside, on or outside the circle (x 3)! + (y + 5)! = 81? Explain how you determined your answer Based on the equation, the center is at ( 1, 2) and the radius is 5 The center of the circle is (3, 5) and the radius is 9 So the distance from the center to the point in question must be compared with 9 d = (9 3) 2 + ( 12 ( 5)) 2 d = (6) 2 + ( 7) 2 d = d = This distance is a little more than 9 units So the point lies outside the circle You Try: Complete the square to find the center and radius of the circle whose equation is x 2 + y 2 4x +8y +11= 0 You Try: Given the endpoints of the diameter of a circle, (10, 5) and ( 2, 9)find the equation of the circle Find the center of the circle using the midpoint formula: x 2 + y 2 4x +8y +11= 0 " $ # 10 + ( 2), 2 5+ ( 9) 2 % " ' $ & # 8 2, 14 2 % ' 4, 7 & ( ) x 2 + y 2 4x +8y = 11 x 2 4x + y 2 +8y = 11 Find the radius using one of the points on the circle and the center: x 2 4x + ( )+ y 2 +8y + ( ) = 11 x 2 4x + ( 2) 2 + y 2 +8y + (4) 2 = 11+ ( 2) 2 + (4) 2 (x 2) 2 + (y + 4) 2 = (x 2) 2 + (y + 4) 2 = 9 d = (10 4) 2 + ( 5 ( 7)) 2 d = (6) 2 + (2) 2 d = d = 40 The equation is (x 4)! + (y + 7)! = 40 GGPE4 GGPE1 Page 12 of 12 MCC@WCCUSD 01/23/15