NAME DATE PERIOD. Find the perimeter and area of each parallelogram. Round to the nearest tenth if necessary. 4 ft. 22 in. 45.
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1 - Skills Practice Area of Parallelograms Find the perimeter and area of each parallelogram Round to the nearest tenth if necessary 0 cm 0 0 cm 4 ft 55 ft 0 4 yd 4 7 yd 45 in 45 in Lesson m 5 km 9 km Find the area of each figure 7 4 CRDINATE GEMETRY Given the coordinates of the vertices of a quadrilateral, determine whether it is a square, a rectangle, or a parallelogram Then find the area of the quadrilateral 9 A(4, ), B(, ), C(, ), 0 P(, ), Q(, ), R(, ), D(4, ) S(, ) D(5, ), E(7, ), F(4, 4), R(, ), S(4, 0), T(, 0), G(, 4) U(0, ) Glencoe/McGraw-Hill Glencoe Geometry
2 - Reading to Learn Mathematics Areas of Triangles, Trapezoids, and Rhombi Pre-Activity How is the area of a triangle related to beach umbrellas? Read the introduction to Lesson - at the top of page 0 in your tetbook Classify the polygons in the panels of the beach umbrella Reading the Lesson Match each area formula from the first column with the corresponding polygon in the second column a A w i triangle b A d d ii parallelogram c A s iii trapezoid d A h(b b ) iv rhombus e A bh v square f A bh vi rectangle Determine whether each statement is always, sometimes, or never true In each case, eplain your reasoning a The area of a square is half the product of its diagonals Lesson - b The area of a triangle is half the product of two of its sides c You can find the area of a rectangle by multiplying base times height d You can find the area of a rectangle by multiplying the lengths of any two of its sides e The area of a trapezoid is the product of its height and the sum of the bases f The square of the length of a side of a square is equal to half the product of its diagonals Helping You Remember A good way to remember a new geometric formula is to state it in words Write a short sentence that tells how to find the area of a trapezoid in a way that is easy to remember Glencoe/McGraw-Hill Glencoe Geometry
3 - Enrichment Areas of Similar Triangles You have learned that if two triangles are similar, the ratio of the lengths of corresponding altitudes is equal to the ratio of the lengths of a pair of corresponding sides However, there is a different relationship between the areas of the two triangles Theorem If two triangles are similar, the ratio of their areas is the square of the ratio of the lengths of a pair of corresponding sides Triangle II is k times larger than Triangle I Thus, its base is k times as large as that of Triangle I and its height is k times as large as that of Triangle I s ide of side o II f I k b b or k a rea of II k bh or k area of I bh h b Triangle I area I bh kh kb Triangle II area II (kb)(kh) k bh Solve DEF GHJ, HJ, and EF In the figure below, PQ BC The area of The area of GHJ is 40 Find the area ABC is 7 Find the area of APQ of DEF D A 4 E F P Q G H J B C Two similar triangles have areas of and The length of a side of the smaller triangle is 0 feet Find the length of the corresponding side of the larger triangle 4 Find the ratio of the areas of two similar triangles if the lengths of two corresponding sides of the triangles are centimeters and 5 centimeters Glencoe/McGraw-Hill Glencoe Geometry
4 - Skills Practice Areas of Regular Polygons and Circles Find the area of each regular polygon Round to the nearest tenth a pentagon with a perimeter of 45 feet a heagon with a side length of 4 inches a nonagon with a side length of meters 4 a triangle with a perimeter of 54 centimeters Find the area of each circle Round to the nearest tenth 5 a circle with a radius of yards a circle with a diameter of millimeters Find the area of each shaded region Assume that all polygons are regular Round to the nearest tenth 7 4 m Lesson - in m ft 5 cm Glencoe/McGraw-Hill 5 Glencoe Geometry
5 -4 Skills Practice Areas of Irregular Figures Find the area of each figure Round to the nearest tenth if necessary y C(, ) y P(0, 7) R(, 7) B(, 5) A(0, 0) D(, 5) E(, 0) Q(, 5) T(, ) U(0, 0) S(, 0) 7 y J(, 4) K(5, 4) E(, 4) F(, 4) y L(, 0) M(, ) D(, ) H(, ) G(4, ) Lesson -4 Glencoe/McGraw-Hill Glencoe Geometry
6 -4 Reading to Learn Mathematics Areas of Irregular Figures Pre-Activity How do windsurfers use area? Read the introduction to Lesson -4 at the top of page 7 in your tetbook How do you think the areas of the figures outlined in the picture of the sail are related? Reading the Lesson Use dashed segments to show how each figure can be subdivided into figures for which you have learned area formulas Name the smaller figures that you have formed as specifically as possible and indicate whether any of them are congruent to each other a b c In the figure, B is the midpoint of ABC Complete the following steps to derive a formula for the area of the shaded region in terms of the radius r of the circle The area of circle P is mabc mab mbc because AB BC because Therefore, ABC is a(n) because triangle AC, so AB and BC The area of ABC is Therefore, the area of the shaded region is given by B A C P Lesson -4 A Helping You Remember Rolando is having trouble remembering when to subtract an area when finding the area of an irregular figure How can you help him remember? Glencoe/McGraw-Hill Glencoe Geometry
7 -5 Study Guide and Intervention Geometric Probability Geometric Probability The probability that a point in a figure will lie in a particular part of the figure can be calculated by dividing the area of the part of the figure by the area of the entire figure The quotient is called the geometric probability for the part of the figure If a point in region A is chosen at random, then the probability P(B) that the point is in region B, which is in the interior of region A, is P(B) a rea of region B area of region A Eample Darts are thrown at a circular dartboard If a dart hits the board, what is the probability that the dart lands in the bull s-eye? Area of bull s-eye: A () or 4 Area of entire dartboard: A (0) or 00 The probability of landing in the bull s-eye is area of bull s-eye area of dartboard 4 00 or in in 4 in Eercises Find the probability that a point chosen at random lies in the shaded region Round to the nearest hundredth if necessary cm cm cm Lesson -5 Glencoe/McGraw-Hill 5 Glencoe Geometry
8 -5 Sectors and Segments of Circles A sector of a circle is a region of a circle bounded by a central angle and its intercepted arc A segment of a circle is bounded by a chord and its arc Geometric probability problems sometimes involve sectors or segments of circles If a sector of a circle has an area of A square units, a central angle measuring N, and a radius of r units, then A N 0 r Eample A regular heagon is inscribed in a circle with diameter Find the probability that a point chosen at random in the circle lies in the shaded region The area of the shaded segment is the area of sector AF the area of AF N Area of sector AF 0 r 0 ( 0 ) Area of AF bh ()() 9 The shaded area is 9 or about The probability is ar ea of se area of gment circle or about 00 Eercises Study Guide and Intervention (continued) Geometric Probability Find the probability that a point in the circle chosen at random lies in the shaded region Round to the nearest hundredth A B F C E D segment sector cm in Glencoe/McGraw-Hill Glencoe Geometry
9 Answers (Lesson -) - Skills Practice Area of Parallelograms Glencoe/McGraw-Hill Glencoe Geometry Answers Lesson - Find the perimeter and area of each parallelogram Round to the nearest tenth if necessary 0 cm 55 ft 0 0 cm 4 ft 0 00 cm, 59 cm 9 ft, 9 ft 4 yd 4 in in 7 yd yd, 9 yd 9 in, 4045 in 5 4 m 5 km 9 km m, m 55 km, 5 km Find the area of each figure units 5 units CRDINATE GEMETRY Given the coordinates of the vertices of a quadrilateral, determine whether it is a square, a rectangle, or a parallelogram Then find the area of the quadrilateral 9 A(4, ), B(, ), C(, ), 0 P(, ), Q(, ), R(, ), D(4, ) S(, ) square, 9 units rectangle, 4 units D(5, ), E(7, ), F(4, 4), R(, ), S(4, 0), T(, 0), G(, 4) U(0, ) parallelogram, 0 units parallelogram, 5 units - Practice (Average) Area of Parallelograms Find the perimeter and area of each parallelogram Round to the nearest tenth if necessary 5 m cm 0 in 0 45 m 0 cm 45 m, 47 m cm, 5 cm 4 in, 50 in Find the area of each figure units 5 units CRDINATE GEMETRY Given the coordinates of the vertices of a quadrilateral, determine whether it is a square, a rectangle, or a parallelogram Then find the area of the quadrilateral C(4, ), D(4, ), F(, ), G(, ) 7 W(, ), X(, ), Y(, ), Z(, ) rectangle, 5 units parallelogram, units M(0, 4), N(4, ), (, ), P(, 0) 9 P(5, ), Q(4, ), R(5, 5), S(4, 5) square, 0 units parallelogram, 7 units FRAMING For Eercises 0, use the following information A rectangular poster measures 4 inches by inches A frame shop fitted the poster with a half-inch mat border 0 Find the area of the poster 09 in Find the area of the mat border 9 in Suppose the wall is marked where the poster will hang The marked area includes an additional -inch space around the poster and frame Find the total wall area that has been marked for the poster 47 in Glencoe/McGraw-Hill 4 Glencoe Geometry Glencoe/McGraw-Hill A Glencoe Geometry
10 - Reading to Learn Mathematics Areas of Triangles, Trapezoids, and Rhombi Glencoe/McGraw-Hill Glencoe Geometry Answers Lesson - Pre-Activity How is the area of a triangle related to beach umbrellas? Read the introduction to Lesson - at the top of page 0 in your tetbook Classify the polygons in the panels of the beach umbrella Isosceles triangles and isosceles trapezoids Reading the Lesson Match each area formula from the first column with the corresponding polygon in the second column a A w vi i triangle b A d d iv ii parallelogram c A s v iii trapezoid d A h(b b ) iii iv rhombus e A bh i v square f A bh ii vi rectangle Determine whether each statement is always, sometimes, or never true In each case, eplain your reasoning For eplanations, sample answers are given a The area of a square is half the product of its diagonals Always; a square is a rhombus, so you can use the rhombus formula b The area of a triangle is half the product of two of its sides Sometimes; this is true only for a right triangle c You can find the area of a rectangle by multiplying base times height Always; a rectangle is a parallelogram, so you can use the parallelogram formula If the length of a rectangle is used as the base, then the width is the height d You can find the area of a rectangle by multiplying the lengths of any two of its sides Sometimes; this is true only for a square therwise, you must use two consecutive sides, not any two sides e The area of a trapezoid is the product of its height and the sum of the bases Never; the area is one-half the product of its height and the sum of the bases f The square of the length of a side of a square is equal to half the product of its diagonals Always; a square is a rhombus, so the formulas for a square and a rhombus must give the same answer whenever the rhombus is a square Helping You Remember A good way to remember a new geometric formula is to state it in words Write a short sentence that tells how to find the area of a trapezoid in a way that is easy to remember Sample answer: Average the lengths of the bases and multiply by the height Areas of Similar Triangles You have learned that if two triangles are similar, the ratio of the lengths of corresponding altitudes is equal to the ratio of the lengths of a pair of corresponding sides However, there is a different relationship between the areas of the two triangles Theorem If two triangles are similar, the ratio of their areas is the square of the ratio of the lengths of a pair of corresponding sides Triangle II is k times larger than Triangle I Thus, its base is k times as large as that of Triangle I and its height is k times as large as that of Triangle I s s a a d d e e o f o f I I I r e a r e a k b o f o f I I I b k or or k Triangle I area I bh Triangle II area II (kb)(kh) k bh Solve DEF GHJ, HJ, and EF In the figure below, PQ BC The area of The area of GHJ is 40 Find the area ABC is 7 Find the area of APQ of DEF 0 D E F G P A 4 Q H J B C Two similar triangles have areas of and The length of a side of the smaller triangle is 0 feet Find the length of the corresponding side of the larger triangle 5 ft 4 Find the ratio of the areas of two similar triangles if the lengths of two corresponding sides of the triangles are centimeters and 5 centimeters 9 5 Glencoe/McGraw-Hill Glencoe Geometry i i Answers (Lesson -) - Enrichment h kh k bh bh b kb Glencoe/McGraw-Hill A7 Glencoe Geometry
11 Answers (Lesson -) - Skills Practice Areas of Regular Polygons and Circles Glencoe/McGraw-Hill 5 Glencoe Geometry Answers Lesson - Find the area of each regular polygon Round to the nearest tenth a pentagon with a perimeter of 45 feet 94 ft a heagon with a side length of 4 inches 4 in a nonagon with a side length of meters 95 m 4 a triangle with a perimeter of 54 centimeters 40 cm Find the area of each circle Round to the nearest tenth 5 a circle with a radius of yards yd a circle with a diameter of millimeters 545 mm Find the area of each shaded region Assume that all polygons are regular Round to the nearest tenth 7 4 m in m in 94 m ft 5 cm 9 ft 5 cm - Practice (Average) Areas of Regular Polygons and Circles Find the area of each regular polygon Round to the nearest tenth a nonagon with a perimeter of 7 millimeters 0447 mm an octagon with a perimeter of 9 yards 95 yd Find the area of each circle Round to the nearest tenth a circle with a diameter of feet 509 ft 4 a circle with a circumference of kilometers km Find the area of each shaded region Assume that all polygons are regular Round to the nearest tenth 5 cm 44 in 44 cm 57 in 7 5 ft 9 m 97 ft 4 m DISPLAYS For Eercises 9 and 0, use the following information A display case in a jewelry store has a base in the shape of a regular octagon The length of each side of the base is 0 inches The owners of the store plan to cover the base in black velvet 9 Find the area of the base of the display case about 4 in 0 Find the number of square yards of fabric needed to cover the base about 07 yd Glencoe/McGraw-Hill Glencoe Geometry Glencoe/McGraw-Hill A9 Glencoe Geometry
12 Answers (Lesson -4) Lesson -4-4 Skills Practice Areas of Irregular Figures Find the area of each figure Round to the nearest tenth if necessary units 40 units units 7 units 0 5 y C(, ) y P(0, 7) R(, 7) B(, 5) D(, 5) Q(, 5) T(, ) E(, 0) U(0, 0) S(, 0) A(0, 0) 40 units 0 units 7 y y J(, 4) K(5, 4) E(, 4) F(, 4) G(4, ) L(, 0) M(, ) D(, ) H(, ) 55 units 0 units Glencoe/McGraw-Hill Glencoe Geometry -4 Practice (Average) Areas of Irregular Figures Find the area of each figure Round to the nearest tenth if necessary units 9 units units 954 units 5 y y D(, 5) Q(, ) E(4, 4) B(, ) C(, 4) P(, ) R(4, ) A(, 0) F(4, 0) T(, ) S(, 0) 05 units units LANDSCAPING For Eercises 7 and, use the following information ne of the displays at a botanical garden is a koi pond with a walkway around it The figure shows the dimensions of the pond and the walkway 7 ft 5 ft ft 5 ft 7 Find the area of the pond to the nearest tenth 95 ft Find the area of the walkway to the nearest tenth 57 ft Glencoe/McGraw-Hill Glencoe Geometry Glencoe/McGraw-Hill A Glencoe Geometry
13 -4 Reading to Learn Mathematics Areas of Irregular Figures Glencoe/McGraw-Hill Glencoe Geometry Answers Lesson -4 Pre-Activity How do windsurfers use area? Read the introduction to Lesson -4 at the top of page 7 in your tetbook How do you think the areas of the figures outlined in the picture of the sail are related? Sample answer: The areas get smaller as you move further up the sail The area of the triangle is smaller than the area of any of the trapezoids Reading the Lesson Use dashed segments to show how each figure can be subdivided into figures for which you have learned area formulas Name the smaller figures that you have formed as specifically as possible and indicate whether any of them are congruent to each other Sample answers are given a b c rectangle and square and rectangle and isosceles triangle two congruent two congruent isosceles triangles semicircles In the figure, B is the midpoint of ABC Complete the following steps to derive a formula for the area of the shaded region in terms of the radius r of the circle The area of circle P is r A mabc 90 because Sample answer: It is an inscribed angle that intercepts a semicircle mab mbc because Sample answer: B is the midpoint of ABC (definition of midpoint) AB BC because Sample answer: If two minor arcs of a circle are congruent, their corresponding chords are congruent Therefore, ABC is a(n) isosceles right or triangle B C P r or r r or r AC r, so AB and BC r r The area of ABC is r Therefore, the area of the shaded region is given by A r r r Helping You Remember Rolando is having trouble remembering when to subtract an area when finding the area of an irregular figure How can you help him remember? Sample answer: Subtract when there is an indentation, or a hole in the figure Aerial Surveyors and Area Many land regions have irregular shapes Aerial surveyors often use coordinates when finding areas of such regions The coordinate method described in the steps below can be used to find the area of any polygonal region Study how this method is used to find the area of the region at the right Step List the ordered pairs for the vertices in counter-clockwise order, repeating the first ordered pair at the bottom of the list Step Find D, the sum of the downward diagonal products (from left to right) D (5 5) ( ) ( ) ( 7) 5 4 or 75 Step Find U, the sum of the upward diagonal products (from left to right) U ( 7) ( 5) ( ) (5 ) or 45 Step 4 Use the formula A (D U) to find the area A (D U) (75 45) (0) or 5 The area is 5 square units Count the number of square units enclosed by the polygon Does this result seem reasonable? Use the coordinate method to find the area of each region in square units y y y 0 units 4 units 4 units Glencoe/McGraw-Hill 4 Glencoe Geometry Answers (Lesson -4) -4 Enrichment y (, 5) (5, 7) (, ) (, ) (5, 7) (, 5) (, ) (, ) (5, 7) Glencoe/McGraw-Hill A Glencoe Geometry
14 Lesson -5-5 Study Guide and Intervention Geometric Probability Geometric Probability The probability that a point in a figure will lie in a particular part of the figure can be calculated by dividing the area of the part of the figure by the area of the entire figure The quotient is called the geometric probability for the part of the figure If a point in region A is chosen at random, then the probability P(B) that the point is in region B, which is in the interior of region A, is P(B) a e a o e a e a o g o n B e g o n A Eample Darts are thrown at a circular dartboard If a dart hits the board, what is the probability that the dart lands in the bull s-eye? Area of bull s-eye: A () or 4 Area of entire dartboard: A (0) or 00 The probability of landing in the bull s-eye is area of bull s-eye area of dartboard or in in 4 in Eercises Find the probability that a point chosen at random lies in the shaded region Round to the nearest hundredth if necessary cm cm cm Glencoe/McGraw-Hill 5 Glencoe Geometry -5 Study Guide and Intervention (continued) Geometric Probability Sectors and Segments of Circles A sector of a circle is a region of a circle bounded by a central angle and its intercepted arc A segment of a circle is bounded by a chord and its arc Geometric probability problems sometimes involve sectors or segments of circles segment sector If a sector of a circle has an area of A square units, a central angle N measuring N, and a radius of r units, then A 0 r Eample A regular heagon is inscribed in a circle with diameter Find the probability that a point chosen at random in the circle lies in the shaded region The area of the shaded segment is the B C area of sector AF the area of AF N Area of sector AF 0 r 0 ( 0 ) Area of AF bh A D F E ()() 9 The shaded area is 9 or about The probability is ar a e r a ea of o s e g f c ir m cl e nt e or about 00 Eercises Find the probability that a point in the circle chosen at random lies in the shaded region Round to the nearest hundredth cm 4 in Glencoe/McGraw-Hill Glencoe Geometry r r f r f r i i Answers (Lesson -5) Glencoe/McGraw-Hill A4 Glencoe Geometry
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