Outside the middle school program / regular polygons 15 exercises (including 11 corrected)

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ABCDEFOC DNMCLKBJIAPO a¸oOb¸oOc¸oOd¸oOe¸oOf¸oO 1. General E.5373 Consider the regular hexagon ABCDEF shown opposite, inscribed in the circle C with center O . 1 a Give the measure of angle COD . b What is the measure of angle COE ? c Deduce the measure of angle EAC . Justify your an-swer. 2 a Give, without justification, the measure of angles ACE and CEA . b What is the nature of the triangle ACE ? E.3958 In the figure below : ABCD is a square of side 9 cm ; segments of equal length are coded. 1 Make a full-scale figure. 2 a Calculate JK . b Is the octagon IJKLMNOP a regular octagon? Jus-tify the answer. c Calculate the area of the octagon IJKLMNOP . 3 The diagonals of the square ABCD intersect at S . a Draw on the full-size figure the circle with center S and diameter 9 cm . b Does the disk of center S and diameter 9 cm have an area greater than the area of the octagon? Justify the answer. 2. Properties of regular polygons E.3957 Consider the figure below showing six reg-ular polygons with the point O as their center : 1 Name each of these six regular polygons. 2 On each of these polygons, is represented an angle hav-ing for center the point O and connecting two consecutive vertices of the regular polygon. In each case, determine the measure of the angle ¸ . 3. Regular polygons and inscribed angles https://chingmath.fr chapExoCorrec/5373 sacados/5373 ABCDEFOC chapExoCorrec/3958 sacados/3958 DNMCLKBJIAPO sacados/3957 a¸oOb¸oOc¸oOd¸oOe¸oOf¸oO
ABCDEFO¸˛‚‹ ABCDEFOH ABCDEFGHIO E.5715 Opposite is a regular hexagon. Determine the measure of the angles coded on the fig-ure. 4. Regular polygons and trigonometry E.4025 Consider the regular hexagon ABCDEF of center O shown below : The radius of the circle has measure 4 cm . Point H is the height from vertex O in triangle OAB . 1 a Determine the measure of the angle AOB . Justify your approach. b Deduce the measure of segment [ AB ] . c Determine the measure of the angle AOH . Justify your approach. 2 Determine the perimeter of the hexagon ABCDEF . 3 The measurements below will be given to the hundredth of a square centimeter. a Determine the area of triangle OAB . b Deduce the area of this hexagon. E.865 ABCDEFGHI is a regular 9-sided poly-gon (called an enneagon) , O is its center and its circumscribed circle has radius 5 cm . 1 What condition must a polygon inscribed in a circle sat-isfy to be regular? 2 a What is the value of the angle AOB ? b Let’s note M the middle of segment [ AB ] . Calculate the length AM rounded to the nearest millimetre. c Give the measure of the perimeter of the enneagon to the nearest millimetre. 3 Explain why the triangle ADG is an equilateral triangle. E.864 Consider the regular octagon ABCDDEFGH . Note O the center of the polygon and C its circumscribed circle. The radius of the circle C is 4 cm . 1 In the octagon ABCDEFGH , give the measure of an angle at the center connecting two of its consecutive ver-tices. 2 Construct to true size the regular octagon ABCDEFGH . 3 a Note I the middle of the segment. Determine the measure of segment [ IA ] to the nearest millimetre. b Deduce the perimeter of the octagon ABCDEFGH to the nearest millimetre. https://chingmath.fr sacados/5715 ABCDEFO¸˛‚‹ chapExoCorrec/4025 sacados/4025 ABCDEFOH chapExoCorrec/865 sacados/865 ABCDEFGHIO chapExoCorrec/864 sacados/864
OABCDEM ABCDEFOH CRSTUMNPQIK CRMNPQIK E.5673 The Pentagon is a building housing the US Department of Defense. It has the shape of a regular pentagon inscribed in a circle of radius OA =238 m . It is represented by the diagram opposite. 1 Calculate the measure of the angle AOB . 2 The height from O in the triangle AOB intersects the side [ AB ] at the point M : a Justify that ( OM ) is also the bisector of AOB and the perpendicular bisector of [ AB ] . b Prove that [ AM ] measures approximately 140 m . c Deduce the measure of the Pentagon’s perimeter, rounded to the nearest centimeter. 5. Regular polygons, scale and trigonometry E.5369 The diagram opposite shows a regular hexagon ABCDEF of 96 m perimeter. It is inscribed in a circle with center O . Seg-ment [ OH ] is a height of trian-gle OBH . 1 Justify that triangle OAB is an equilateral triangle. 2 Calculate the length OH , expressed in m . Give the rounding to the nearest centimeter. 3 Use this result to calculate the area of the triangle OBA , rounded to the nearest tenth of m 2 . 4 Deduce the area of a regular hexagon of 96 m perimeter, rounded to the nearest m 2 . E.5371 Consider the regular oc-tagon MNPQRSTU représenté in reduction shown opposite, where the segment [ MN ] mesure 12 m en true size. The point K represents the foot of the height from I . 1 Give the measure of the angle MNI . 2 We wish to represent in the frame below, the triangle IMN à the scale 1 = 4 000 a Give the measurement of the segment representing the [ MN ] side. b Effect the representation of the triangle IMN in the frame by adding the height [ IK ] . 3 Determine the length of the height [ IK ] , rounded to the nearest metre. E.5372 Consider the regular pentagon MNPQR inscrit in the circle C de center I représenté in re-duction below. against where the segment [ MN ] measures 12 m in true size. The point K represents the foot of the height from I . 1 Determine the measure of the angle MNI . 2 We wish to represent in the frame below, the triangle IMN à the scale 1 = 4 000 a Give the measurement of the segment representing the [ MN ] side. b Effect the representation of the triangle IMN in the frame by adding the height [ IK ] . 3 Determine the length of the height [ IK ] , rounded to the nearest metre. https://chingmath.fr sacados/5673 OABCDEM chapExoCorrec/5369 sacados/5369 ABCDEFOH chapExoCorrec/5371 sacados/5371 CRSTUMNPQIK chapExoCorrec/5372 sacados/5372 CRMNPQIK
basehauteur ABCDEFOH RSTUMNPQIK 6. The patent problem E.5370 Remember that the area of a tri-angle is calculated using the formula : base × height 2 Remy has 96 m of wire fencing with which he wants to build an enclosure for his pony. He is trying to decide what shape to give his enclosure so that it has the largest possible surface area. All parts are independent Part 1 His first idea is to make a rectangle with the 96 m of wire fencing. Calculate the length and width of this rectangle, knowing that : the length is twice the width ; its perimeter is 96 m . Calculate the area of this rectangle with a perimeter of 96 m . Part 2 His second idea is to make a square. Calculate the area of a square with a perimeter of 96 m . Part 3 His third idea is to make a reg-ular hexagon. The freehand diagram opposite represents a regular hexagon ABCDEF de 96 m perimeter. It is inscribed in a circle of cen-ter O and radius 16 m . The seg-ment [ OH ] is a height of the equilateral triangle OBH . 1 Calculate the length OH , expressed in m et rounded to the nearest centimetre. 2 Use this result to calculate the area of the triangle OBA , rounded to the nearest tenth of a metre-square. 3 Deduce the rounding to unity of the area of a regular hexagon of 96 m perimeter. Part 4 His fourth idea is to create a regular octagon with a perime-ter of 96 m . The figure opposite shows the plan drawn up by Rémy. This octagon is inscribed in a circle with center I . The seg-ment [ IK ] is the height of the isosceles triangle IMN . 1 Verify that MN =12 m in reality. 2 Using a scale of 1 cm for 4 m , draw triangle IMN in the box below, then point K . Leave all construction lines visible. 3 Measure the length IK on your plan. How many meters does this represent in reality? 4 Deduce the area of the triangle MIN , then, using this value, calculate the area of a regular octagon with a perimeter of 96 m . Part 5 Through research, Remy noticed that the area of a regular polygon of 96 m perimeter seems to increase when you in-crease the number of its sides. He imagines that a circular enclosure would perhaps have an even larger surface area. 1 What radius must be taken to have a disk with perimeter 96 m ? This value will be rounded to the hundredth of a centimetre. 2 Determine the area of a disk with perimeter 96 m , rounded to the nearest metre. 7. Drawing polygons E.3955 Using only the compass and a straight-edge, draw the equilateral triangle ABC whose vertex A and center O are shown below : https://chingmath.fr chapExoCorrec/5370 sacados/5370 basehauteur ABCDEFOH RSTUMNPQIK chapExoCorrec/3955 sacados/3955
AO EO 8. Drawing polygons E.3954 Using only the compass and a straight-edge, draw the regular hexagon ABCDDEF whose center O and radius [ OE ] are shown below : 9. Unclassified financial years E.6287 Wind turbines are built to have the same angle measurement between each of their blades. 1 A wind turbine has three blades. What is the measure of the angle between two of its blades? 2 To reduce the noise caused by wind turbines, the number of blades must be increased. The mast of a six-bladed wind turbine is represented be-low by the segment [ AB ] . Taking the point A as the cen- ter of the blades, complete the construction with 5 cm blades. https://chingmath.fr AO chapExoCorrec/3954 sacados/3954 EO sacados/6287
AB 35mA80moreilles1;80mMâtUne paleCentre des palesBCSolLa ∏gure n’est pas à l’échelle 3 It is estimated that at 80 m from the center of a wind turbine’s blades the sound level is just sufficient for the noise it produces to be heard. A hiker whose ears are 1.80 m from the ground moves towards a wind turbine whose mast is 35 m high. He stops as soon as he hears the noise it makes (see diagram below) . How far from the wind turbine mast is (distance BC ) ? Round the result to the nearest whole number. https://chingmath.fr AB 35mA80moreilles1;80mMâtUne paleCentre des palesBCSolLa ∏gure n’est pas à l’échelle