Name ate lass eteaching Nets INV 5 You have worked with two and three-dimensional figures before. Now ou ll work with nets, which are - representations of 3- figures. Making a 3- Figure from a Net A net is a two-dimensional representation of a 3- figure. ach shape in the net represents one face of the figure. folding along the edges of the figures, the net can be made into the 3- figure it represents. ample: Figure 1 shows a net of a cube. ach square represents one face of the cube. In figure the squares are being folded upwards so that it becomes the cube in figure 3. Figure 1 Figure Figure 3 5 1 3 1 5 3 1 5 3 op the nets given in problems 1 3. Form the 3- figures the represent. tate the name of each figure. 1 3 1. tep One: Fold the triangles up along the edges of the rectangle. 5 tep Two: The figure has two triangular bases and three rectangular faces. The figure is a prism. 1 3 5. pentagonal prism 3. an ou form a rectangular prism with the net in figure 5? If so, form it. If not, eplain wh. A rectangular prism cannot be formed with this net, because there is no wa to give the shape a side parallel to side. 1 3 5 aon. All rights reserved. 109 aon Geometr
eteaching continued INV 5 Making Nets of a 3- Figure A 3- figure with flat faces is called a polhedron. To construct the net of a figure, imagine unfolding it along its edges so that it is a flat shape. ample: The cube in figure 1 unfolds in figure to become the net in figure 3. Figure 1 Figure Figure 3 Most polhedrons have more than one net. an ou think of a second net that represents the same cube as the one represented b figure 1? raw a net for the given polhedron.. raw a possible net for this triangular prism. tep One: Fold the triangles out. tep Two: Fold the rectangles out. tep Three: raw the net. You can verif our answer b re-creating the triangular prism using the net ou ve made. 5. raw two possible nets for this truncated pramid. (Note: the sides are trapezoids and the top and bottom are squares) aon. All rights reserved. 110 aon Geometr
Name ate lass eteaching Properties of Isosceles and quilateral Triangles 51 You have worked with different tpes of triangles. Now ou will work specificall with isosceles and equilateral triangles. Isosceles Triangles Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite the sides are congruent. T If T, then T. onverse of Isosceles Triangle Theorem L It two angles of a triangle are congruent, then the sides opposite those angles are congruent. N M If N M, then LN LM. You can use these theorems to find angle measures in isosceles triangles. Find m in F. m m Isosceles Triangle Theorem 5 (3 + 1) 5 3 1 ubstitute. 1 ubtract 3 from both sides. 7 ivide both sides b. F m 3(7) 1 35 omplete the statements to find m A and m. 1. m A m m 180 78 180 78 180 10 51 Find each angle measure.. m Q 7 3. m H 7. m M 0 Q G 8 H A 78 L P 8 ( + 18) J N ( + 30) M aon. All rights reserved. 111 aon Geometr
eteaching continued 51 quilateral Triangles quilateral Triangle orollar If a triangle is equiangular, then it is equilateral. If A, then A A. A You can use this theorem to find values in equilateral triangles. Find in TV. 7 08 7 5 ubtract from both sides. 8 ivide both sides b 7. (7 + ) V T omplete the statements to find the value. 5. 0 33 ( - ) Find each value. F. n 1 7. VT 18 8. MN 13 Q M 9r + 10 + 1 5n V 5r + 8 T L N 9. A triangle is equiangular and has a perimeter of 37. inches. etermine the lenght of each side. 1. inches aon. All rights reserved. 11 aon Geometr
Name ate lass eteaching Properties of ectangles, hombuses, and quares 5 You have worked with special triangles. Now ou will work with properties of parallelograms. Properties of pecial Parallelograms Properties of ectangles The diagonals of a rectangle are Properties of a hombus The diagonals of a rhombus are perpendicular. congruent. H J ach diagonal of a rhombus bisects opposite GJ HK G K angles. Q T Q QT A is a rectangle. Find the length of and of. 5 Opposite sides of a rectangle are congruent. iagonals of a parallelogram bisect each other. It is given that.5. Therefore,.5 also. egment Addition Postulate.5.5 13 inches ubstitution 5 in. A Q 1 in. T.5 in. omplete the statements. 1. If m A 18, then m A. iagonals of a rhombus bisect opposite angles. Use the sum of the angles in a quadrilateral to find and m. 18 18 30 5 30 A 10 5 m 5 m KLMN is a rhombus. Find each measure. L 3 +. KL 8 3. m MNK 50 9º + 0 K ( + 5)º N aon. All rights reserved. 113 aon Geometr M
eteaching continued 5 Using Properties of Parallelograms HJKL is a parallelogram. ecide what tpe of parallelogram it is b using the properties of rectangles and rhombuses. tep 1: etermine if HK JL. Use the distance formula. HK J(, ) ( 0) ( ) 10 JK ( ) (0 ) K(, ) 10 HK JL 10, so HK 3 H(0, ) JL, and HJKL is a rectangle. tep : etermine if HK JL Find the slope. slope of HK 1 slope of JL 0 3 0 3 ince the product of the slopes is 1, HK JL, and HJKL is a rhombus. tep 3: ince HJKL is both a rectangle and a rhombus, it is a square. 3 L(, 0). etermine if A. A (8 0) (1 ( 1)) 8 A, so A. (5 3) ( ) A is a rectangle. 8 5. etermine if slope of A. Find the slope. 5 3 8 1 The product of the slopes is 1, so A. slope of A - - 8 0 1 ( 1) 8 8 - A is a rhombus. ince A is both a rectangle and a rhombus, it is a square.. Find the lengths and the slopes of the diagonals. etermine what tpe of parallelogram PQ is. P Q P 5, and Q 5. PQ is not a rectangle. lope of P 1 PQ is a rhombus., and slope of Q. aon. All rights reserved. 11 aon Geometr
Name ate lass eteaching 5-5 -90 ight Triangles 53 You have worked with right triangles. Now ou will work with a particular kind of right triangle. pecial ight Triangle ide Lengths in 5-5 -90 Triangles ample In a 5-5 -90 right triangle, both legs are congruent, and the length of 5 the hpotenuse is the length of a leg multiplied b 8 8. 8 3 5 5 3 5 3 Find the value of in FG The hpotenuse has length 10. F 10 G 10 Hpotenuse is times the length of a leg. 10 ivide both sides b. 5 ationalize the denominator. omplete the statements to find the value of. 1. The leg is given. It is 17. The hpotenuse is the length of the leg times. 17 5 5 The hpotenuse is 17, so 17 Find the value of. 5 5 5 8 5. 5 3. aon. All rights reserved. 115 aon Geometr
eteaching continued 53 Finding the Perimeter of a 5-5 -90 ight Triangle Find the perimeter of the triangle. tep 1: etermine the length of each leg. The legs are each 5 feet long. 5 tep : etermine the length of the hpotenuse. The hpotenuse is times the length of a leg, so the hpotenuse is 5 feet. tep 3: The perimeter is the sum of all the sides. P 5 5 5 P 10 5 P 17.07 The perimeter is approimatel 17 feet. 5 feet 5 omplete the statements to determine the perimeter of the triangle.. etermine the length of the hpotenuse. The hpotenuse is 3. etermine the length of each leg. The legs are each 3 feet long. The perimeter is the sum of all the sides. P 3 3 3 P 3 P 10. The perimeter is approimatel 10 feet. 5 3 ft 5 etermine the perimeter of each 5-5 -90 right triangle. 5.. 5 5 1 m 7 ft 5 5 3.90 ft 0.97 1 m aon. All rights reserved. 11 aon Geometr
Name ate lass eteaching epresenting olids 5 You have worked with different solids. Now ou will work on drawing different perspectives of solids. rawing in One-Point Perspective In a one-point perspective drawing, non-vertical lines are drawn so that the meet at a vanishing point. ample: raw a triangular prism in one-point perspective. tep 1: tep : tep 1 raw a horizontal line and a vanishing point on the line. raw a triangle below the line. tep From each verte of the triangle, draw dashed segments to the vanishing point. tep 3: tep 3 raw a smaller triangle with vertices on the dashed segments. tep : tep raw the edges of the prism. Use dashed lines for hidden edges. rase segments that are not part of the prism. 1. raw a rectangular prism in one-point perspective that has the given rectangle as its front face and the given point as its vanishing point. tep 1: The rectangle is the front face of the solid. The given point is the vanishing point. tep : raw a dashed line from the vanishing point to each verte of the rectangle. tep 3: raw a smaller rectangle with vertices on the dashed segments. tep : raw the edges of the prism. Use dashed lines for hidden edges. rase segments that are not part of the prism. aon. All rights reserved. 117 aon Geometr
eteaching continued 5 reating Isometric rawings An isometric drawing is a wa of drawing a solid figure b using isometric dot paper. The drawings can be made b using three aes that intersect to form 10 angles. ample: reate an isometric drawing of a triangular prism. tep 1: raw the three aes on isometric dot paper. tep : Use the intersection of the three aes as the bottom verte of the prism. tep 3: raw the prism so that the edges of the prism are parallel to the three aes.. Follow the steps below to create an isometric drawing of a cube. tep 1: raw the three aes on isometric dot paper. tep : Use the intersection of the three aes as the bottom verte of the cube. tep 3: raw the cube so that the edges of the cube are parallel to the three aes. 3. reate an isometric drawing of a rectangular prism. aon. All rights reserved. 118 aon Geometr
Name ate lass eteaching Triangle Midsegment Theorem 55 You have worked with midpoints. Now ou will work with midsegments of triangles. Triangle Midsegment Theorem The segment joining the midpoints of two sides of a triangle is parallel to, and half the length of, the third side. PQ is the midsegment of LMN. PQ LN, PQ 1 LN L P M O N ample: Find the following measures in A. HJ 35 HJ 1 H J A Midsegement Theorem HJ 1 (1) ubstitute 1 for A. A K HJ olve. A m A JK 1 A Midsegement Theorem HJ A Midsegment Theorem 1 A ubstitute for JK. m A m JH orresponding Angle Theroem 8 A olve. m A 35 ubstitute 35 for m JH. 1 omplete the statements to find the measure. 1. VX. HJ H VX 1 GH WV 1 HJ VX 1 () 7 1 HJ W 7 9 X VX 3 5 HJ G V J Find each measure. 3. T 7. 5. m 8. m 8 8 3 T aon. All rights reserved. 119 aon Geometr
eteaching continued 55 Finding Midpoints of ides of Triangles A midsegment of a triangle joins the midpoints of two sides of the triangle. ver triangle has three midsegments. We can use the Midpoint Formula to find the coordinates of the endpoints of a midsegment. is the midpoint of is the midpoint of is a midsegment of. omplete the statements to find the coordinates of the midsegment. 7. Triangle MNP has vertices M(, 7), N(, 5), and P(, 1). UV is a midsegment of MNP. Find the coordinates of U and V. M(-, 7) N(, 5) U and V are the midpoints of MP and PN. U,, 7 1 V,, 5 1-3 U V 3 P(, -1) ( 1, 3) (3, ) olve. 8. Triangle T has vertices (0, 3), (, 3), and T(, 1). A is the midsegment of T. Find the coordinates of A and. A and are the midpoints of and T. A (1, 0); (, 1) 9. Triangle XYZ has vertices X(3, 0), Y(5, ), and Z(9, ). LM is the midsegment of XYZ. Find the coordinates of L and M. L and M are the midpoints of XY and YZ. L (, 3); M (7, ) 10. Triangle A has vertices A(1, 1), (9, 10), and (7, ). is the midsegment of A. Find the coordinates of and. and are the midpoints of and A. (8, 7); (, 8) aon. All rights reserved. 10 aon Geometr
Name ate lass eteaching 30-0 -90 ight Triangles 5 You have worked with 5-5 -90 right triangles. Now ou will work with 30-0 -90 right triangles. pecial ight Triangle 30-0 -90 Triangle ample In a 30-0 -90 right triangle, the length of the hpotenuse is twice the length of the short leg, and the length of the longer leg is the length of the shorter leg times 3. 5 3 30 10 11 11 3 0 5 0 30 ample Find the values of and in HJK. 1 3 Longer leg shorter leg times 3. J 30 1 3 3 3 ivide both sides b 3. 3 ationalize the denominator. 1 Hpotenuse equals two times the shorter leg. ( 3 ) ubstitute 3 for. H 0 K 8 3 implif. omplete the statements to find the value of and. 1. 18. 3 9 9 3 30 18 0 Find the value of and. 0 0 3 30 30 33 3. 1 3 ; 3. 11 3 ; 3 aon. All rights reserved. 11 aon Geometr
eteaching continued 5 Finding the Perimeter of a 30-0 -90 ight Triangle ample: Find the perimeter of the triangle. tep 1: etermine the length of each leg. The short leg is ft. The longer leg is 3 ft. tep : etermine the length of the hpotenuse. 30 The hpotenuse is times the shorter leg. () ft tep 3: The perimeter is the sum of all the sides. 0 P 3 P 3 P 9. The perimeter is approimatel 9 feet. omplete the statements to determine the perimeter of the triangle. 5. horter Leg: Hpotenuse: Perimeter: 3 3 P 8 3 1 3 3 3 (8 3 ) 3 8 3 1 3 5.5 The perimeter is about feet. etermine the perimeter of each 30-0 -90 triangle. 30 ft 0. 30 7. 30 0 9 ft 0 ft 0 9 ft.58 3 feet 7.3 7 feet aon. All rights reserved. 1 aon Geometr
Name ate lass You have worked with special triangles. Now ou will find perimeter and area with coordinates. Finding Perimeter with oordinates Find the perimeter of rectangle TV with coordinates ( 5, 1), ( 3, 5), T (3, ), and V (1, ). tep 1: Plot the points on a coordinate plane and draw TV. tep : Use the distance formula to calculate the length of each side. Find the length of first. d ( 1 ) ( 1 ) ecause the figure is a rectangle, d TV 5. ( 3 ( 5)) (5 1) () () 0 5 tep 3: Find the length of one of the longer sides. d V (1 ( 5)) ( 1) 3 5 tep : The length of the opposite side, T, is 3 5. tep 5: Perimeter: P T TV V P 5 3 5 5 3 5 eteaching Finding Perimeter and Area with oordinates P 10 5 or approimatel.3 5 5 V 3 T 57 omplete the steps to find the perimeter of the triangle. 1. d (3 ( 1 )) ( ( )) 3 8 5..83 d (5 ( 1)) 0.3 (0 ( )) d (5 3) (0 ) P 5..83.3 1.81 aon. All rights reserved. 13 aon Geometr
eteaching continued 57 stimating Area with oordinates Find the area of the polgon with vertices N (, 1), P( 1, 3), Q (, 3), and (, ). tep 1: ount all the squares completel inside the polgon: 17. tep : stimate the area of the remaining space. Look for triangles. N P Q N P 1 3 5 7 8 9 10 111 13 1 15 117 Q N P 1 3 5 7 8 9 10 111 13 1 15 117 Q Two right triangles under point Q each have an area about 1.5 square units, for a total of 3 square units. Another right triangle b point P has an area of about square units. The right triangle b point N has an area of about 1 square unit. The right triangle at the bottom of the polgon has an area of about 3 square units. There is one remaining area near point N that is almost one whole square unit. tep 3: Add the estimates together to estimate the area of polgon NPQ. A 17 3 1 3 1 7 square units omplete the steps to estimate the area of the polgon.. Find the areas of the different figures in the polgon. The number of complete squares inside the polgon is 0. stimate the remaining area b finding areas of right triangles inside the polgon: 9 square units. A 0 9 9 square units X(-3, ) 0 W(-3, -1) Y(3, 1) Z(, -) 3. Find the areas of the different figures in the polgon. The number of complete squares inside the polgon is 10. stimate the remaining area b finding areas of right triangles inside the polgon: 9.5 square units. A 10 9.5 19.5 square units (-3, 1) 0 T(, 3) U(, 0) V(3, -3) aon. All rights reserved. 1 aon Geometr
Name ate lass eteaching Tangents and ircles, Part 1 58 Tangent Lines and Angle Measures A tangent line is a line that is in the same plane as a circle and intersects the circle at eactl one point. Theorems If a line is tangent to a circle, then the line is perpendicular to a radius drawn to the point of tangenc. A A radius tangent If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle. A is tangent to. Which line is tangent to A at point? Line b is tangent to A at point. If m A 5, find m A. ince A is the radius of the circle, A b and m A 90. Therefore, A is a right triangle. m A m A 90 5 m A 90 m A 90 5 m A 5 a A b If line m is tangent to at point and m T 8, find m T and m T. 1. ince is the radius of the circle and is perpendicular to line m, m T 90. 90 m T m T m l 8 m T 90 m T 90 8 m T Line b is tangent to J at point N, and m NJM 3. T M. Find m JNM. 3. Find m JMN. N 90 58 J b aon. All rights reserved. 15 aon Geometr
eteaching continued 58 Appling elationships of Tangents from an terior Point Theorem If two segments are tangent to a circle from the same eterior point, then the are congruent. F F G G KN and MN are tangent to L from point N. etermine the perimeter of quadrilateral KLMN. ince LM and LK are radii of the same circle, the are congruent. Therefore, LM in. in. K 3 in. ince KN and MN are tangents to the same circle from the same eterior point, the are congruent. L N Therefore, MN 3 in. P KL LM MN KN M 3 3 10 in. The segments are tangent to the circle from the same eterior point. omplete the steps to find the perimeter of the quadrilateral.. A cm; A cm cm; cm cm A cm P A A P P 0 cm The segments are tangent to the circle from the same eterior points. Find the perimeter of each quadrilateral. 5. M 10 in. N. 5 in. K 7 m P 5 m L T 30in. m aon. All rights reserved. 1 aon Geometr
Name ate lass eteaching Finding urface Area and Volumes of Prisms 59 You have worked with tangents and circles. Now ou will find surface area and volume of prisms. urface Area of Prisms The lateral area of a prism is the sum of the areas of all the lateral faces. A lateral face is not a base. The surface area is the total area of all faces. The lateral area of a prism with base perimeter p and height h is L ph. h The surface area of a prism with lateral area L and base area is L. Find the lateral area and the surface area of the rectangular prism. tep 1: The perimeter is p (5) () in. Therefore, L ph ()() 13 in. tep : Find the area of the base. The area of the base is ()(5) 30 in. tep 3: Find the total surface area. L 13 (30) 19 in. The surface area of the prism is 19 square inches. in. lateral face 5 in. in. omplete the steps to find the lateral area and surface area of the prism. 1. lateral area: p (8) ( 7 ) 30m 5 m surface area: area of each base (8)( 7 ) 5 m 8 m 7 m L 150 ( 5 ) m Find the lateral area and surface area of each prism.. 3 ft 3. cm ft 9 ft cm cm L 78 ft ; 150 ft L 10 cm ; 18 cm aon. All rights reserved. 17 aon Geometr
eteaching continued 59 Volume of Prisms The volume of a prism with base area and height h is V h. h Find the volume of the right prism. tep 1: Find the area of the base. The base is a right triangle. The area of the base is 1 (5)(1) 30 m. tep : The height of the prism is h 3 m. The volume of the prism is V h (30)(3) 90 m 3 The volume of the right prism is 90 cubic meters. 5 m 3 m 1 m omplete the steps to find the volume of the prism.. () ( 8 ) 1 in 5. 1 () ( ) 8 d The height of the prism is in. The height of the prism is d. 3 V h ( 1 ) () in V h ( 8 ) () 3 d 3 in. d 8 in. in. d d Find the volume of each prism.. 7. 9 cm 5 in. 1 cm cm 3 in. 8 in. 57 cm 3 0 in 3 aon. All rights reserved. 18 aon Geometr
Name ate lass You have worked with proportional relationships that eist within triangles. Now ou will work with an line that intersects a triangle and that is parallel to a base. Triangle Proportionalit Theorem eteaching Proportionalit Theorems 0 If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionall. XY A o X XA = Y Y X Y A You can use the Triangle Proportionalit Theorem to find lengths of segments in triangles. ample: Find the length of G. G H GF Triangle Proportionalit Theorem HF G 7.5 ubstitute the known values. 5 G(5) (7.5) ross Products Propert 5(G) 5 implif. G 9 ivide both sides b 5. G 7.5 H 5 F omplete the statements to find the length of Q. 1. N Q P Q 1 Q 7 1(7) (Q) N 1 P 7 Q 8 (Q) 1 Q Find the length of each segment.. JN 30. 3. H 5. K 0 M 1 J N 38 L H 9 9 I 15 aon. All rights reserved. 19 aon Geometr
eteaching continued 0 Parallel Lines and Transversals If parallel lines intersect transversals, then the divide the transversals proportionall. A F A F You can use the Triangle Proportionalit Theorem to find lengths of segments. Find the length of KL. JK IH KL J Triangle Proportionalit Theorem 10 HG 10 K ubstitute the known values. KL 8 1 10(1) KL(8) ross Products Propert 10 8(KL) implif. 15 KL ivide both sides b 8. The length of KL is 15. L I 8 H 1 G. A 1 F 1 18 1(18) (1) A 1 F 1 18 88 1() 5. Find the length of segment LM 7. L M 1 N P 10 Q 0 aon. All rights reserved. 130 aon Geometr