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IES Llanes. Sección Bilingüe Dpto. De Matemáticas IES Llanes. Sección bilingüe Dpto. de Matemáticas 1 Index of contents: Plane Geometry……………………………………………………………………………………...........1 Similarity.............................................................................................................12 The right triangle.............................................................................................14 Polyhedra..............................................................................................................17 Solids of revolution...........................................................................................20 Statistics and probability..............................................................................22 Integer numbers...............................................................................................26 Rational numbers...............................................................................................29 Proportionality……………………………………………………………………………………………...31 Algebra…………………………………….........................................................................33 2 3 4 CIRCULAR SHAPES: . Circumference: is a curved line in which every point is at the same distance from a fixed point called the centre. Circle: is the zone inside a circumference. Lines of a circumference and number Pi: . Circular slices: 5 Area of PlaneShapes Triangle Square Area = ½b × h Area = a2 b = base a = length of side h = vertical height Rectangle Parallelogram Area = w × h Area = b × h w = width b = base h = height h = vertical height Trapezoid (US) Circle Trapezium (UK) Area = πr2 Area = ½(a+b) × h Circumference=2πr h = vertical height r = radius Sector Ellipse Area = ½r2θ Area = πab r = radius θ = angle in radians Angles: . Units: degrees ( º ) o A circle is divided into 360 equal degrees, so, 1 circle = 360º o A right angle=90º o Degrees can be divided into minutes: 1º = 60’ o Minutes can be divided into seconds: 1’ = 60’’ . Another unit is the radian; 1 circle = 2 rad 6.28 rad o 1 radian 57º Angles in the circumference: inscribed half-inscribed interior exterior 6 EXERCISES 1. Write the correct word for each definition: a. Lines that never meet are…………….....lines b. A triangle with three equal sides is a ………..….......triangle c. A quadrilateral with four equal sides and two pairs of equal angles is a …….............. d. The perpendicular bisectors of a triangle meet on a point called………….............. e. The heights of a triangle meet on a point called………................... f. A ...................... angle measures more than 180º but less than 360º g. Lines that meet on one point are………….........…..lines h. A triangle with two equal sides is an ………..…............triangle i. A quadrilateral with four equal angles and two pairs of equal sides is a ……............... j. The medians of a triangle meet on a point called ………….......... k. The bisectors of a triangle meet on a point called ………............ 2. Classify these triangles taking into account their sides and their angles: a. Two angles of 35º and 55º b. Two angles of 35º each one c. Two angles of 55º and 65º d. Two angles of 45º each one e. Two angles of 30º and 50º 3. Find the measures of all the unknown angles in these polygons: a. A rhombus with an angle of 130º b. An isosceles trapezium with an angle of 110º c. A rhombus with an angle of 130º d. An isosceles trapezium with an angle of 110º e. A right triangle with an angle of 40º f. An isosceles triangle with the different angle of 120º 7 4. Classify these quadrilateral: a. Four equal sides and for equal angles. b. Only opposite sides are parallel. c. Four eual angles and two pairs of equal sides. d. Four equal sides and two pairs of equal angles. 5. Solve these exercises about area and perimeter: 5.1 Get the diagonal, the perimeter and the área of this rectangle 5.2Get the perimeter and the área of this right trapezium: 5.3 Calculate the perimeter and the área of this polygon and write its name: 5.4Calculate the perimeter and the área of these shapes: 8 6. Mark goes riding his bike around a track of radius 50 m. How far does he go in 1 lap? In 5 laps? In 3 and ½ laps? 7. Answer these questions: . How many diameters does a circumference have? . How many centres? . How many radius? 8. A thread with a length of 50 cm can be rolled around a reel (carrete) 10 times. What is the diameter of the reel? 9. What is the length of an arc with a radius and angle of: . 4 cm and 20º . 5 km and 150º . 12 m and 30º 10. James and Tom want to share a pizza with radius 30 cm. James will eat 3/5 of the pizza. Which part (fraction) of the pizza will be for Tom? How many square cm will each one eat? 11. Convert these angles into single units: . 20º 30’ . 45º 20’ 30’’ . 120º 120’’ 12. Convert these angles into complex units: . 7200’’ . 1354’ . 125’ 13. Calculate: . 26º 30’ 15’’ – (15º 40’ 25’’) . 3 · (35º 35’ 30’’) . (121º 37’ 25’’):5 . 17 h 15 min – (14 h 30 min) . 7· (4h 20min 50s) 9 Similarity 1. Are similar two triangles with these sides? a 6cm, b 9cm, c 5 cm. a'12 cm, b'18 cm, c'10 cm 2. Complete the measures (lengths) of the sides in these triangles to be similar: 3 cm6 cm 2 cm4 cm10 cm 3. What is the height of the tree? Use these data: height and shadow of the statue: 3 m and 4 m. respectively; shadow of the tree: 6 m 4. Draw a segment with a length of 5 cm and divide it into 7 parts. 5. A square has an area of 4 cm2 . What is the area of another square with a side: a)double? b) half? 6. Find similar triangles among these right-angled ones: 63º 41º 27º 45º 49º 47º A C D E 7. The length of a road on B a map with a scale of 1:500 000 is 8 cm. What F is its real length? 8. The distance between Sevilla and Madrid is about 540 km. What is this distance on a map with a scale of 1:150 0001 9. Calculate the distance between Tribeca and City Hall 10 11. There are four sets of similar triangles below. Can you work out the lengths of the sides marked with a question mark 11 Similar polygons have the same shape but different size. a b c d The lengths of their sides satisfy the relationship: a' b' c? d' Their corresponding angles are equal: A = A’, B = B’ , C = C’ , D = D’ d d’ a c a’ c’ b b’ Similar triangles: two triangles are similar if: 3 angles of one triangle are the same as 3 triangles of the other, or 3 pairs of corresponding sides are in the same ratio (proportional), or An angle of one triangle is the same as the angle of the other triangle and the sides containig these angles are in the same ratio. 12 The right triangle 1. Can you construct a triangle with these lengths? If it is possible, what kind of triangle is it? (right triangle, acute triangle, obtuse triangle) a. 3 m, 4m, 7m b. 4 cm, 5 cm, 6 cm c. 3m, 5m, 7m d. 6 cm, 8 cm, 10 cm 2. What is the diagonal of a rectangle with sides of 3 m and 7 m? 3. What is the apothem of a hexagon with sides of 6 cm? Solve these Pythagorean problems: 4. To get from point A to point B you must avoid walking through a pond(charco). To avoid the pond, you must walk 34 metres south and 41metres east. If you walk directly from A to B, how many metres will you walk? Round to the nearest dm. 5. A baseball diamond is a square with sides of 90 feet. What is the shortest distance, to the nearest tenth of a foot, between first base and third base? 6. A suitcase measures 24 inches long and 18 inches high. What is the diagonal length of the suitcase to the nearest tenth? 7. Two joggers run 8 miles north and then 5 miles west. What is the shortest distance, to the nearest tenth of a mile, they must travel to return to their starting point? 13 8. Find the height of an isosceles triangle with base of 48 cm and equal sides of 26 cm. 9. Find the area of a triangle with the following sides: a. 20 cm, 26 cm, 26 cm b. 2 mm, 2mmm (right triangle) c. 20 m, 20 m, 24 m d. 8 cm of hypotenuse 10. What is the height of a equilateral triangle with sides of 10 dm? 11. Calculate c in the figure 12. Calculate a and bin each triangle: a) b) 13. What is the length of the diagonal AB? 14. Calculate the area of this roof: 14 A right triangle (or right-angled triangle, formerly called a rectangled triangle) has one 90° internal angle (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. The other two sides are the legs or catheti (singular: cathetus) of the triangle. Pythagorean theorem:”Thesquare on the hypotenuse is equal to the sum of the squares on the legs”. If we let c be the length of the hypotenuse and a and b be the lengths of the other two sides, the theorem can be expressed as the equation 15 Solid geometry: Polyhedra and solids of revolution 1. Write the names of these polyhedra: 2. Draw the following polyhedra and mark the main elements on them: e. A regular triangular prism. f. A regular hexagonal prism. g. A square pyramid. 3. Draw the net (plane development) for the shapes in exercise 2, and identify the elements. 4. Can you recognize the shapes with these nets? 5. The edges of a cuboid measure 7m, 4m, and 4m; is it a regular or an irregular prism? What is its surface area? And its volume? 6. Get the surface area and the volume of the shapes below: 16 The dihedral angle is the angle between two planes.
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