DrFrostMaths



Year 9 – IMC Geometry ExercisesExercise 1 – Lengths5429250635000[Kangaroo Pink 2008 Q3] Four unit squares are placed edge to edge as shown. What is the length of the line PQ?A 5B 13C 5+2D 5E 135440680317500Two identical squares are inscribed in a semicircle of radius2. Find the length of each square.5370830254000A square of length 2 is inscribed in a semicircle. Find the radius of the semicircle.5172075508000[Kangaroo Pink 2011 Q6] The diagram shows a shape made from a regular hexagon of side one unit, six triangles and six squares. What is the perimeter of the shape?A 6(1+2)B 61+123C 12D 6+32E 95586095762000[IMC 2009 Q7] Four touching circles all have radius 1 and their centres are at the corners of a square. What is the radius of the circle through the points of contact X, Y, Z and T?A 12B 122C 1D 2E 25390795444500[Based on IMC 2015 Q24] In square RSTU a quarter-circle arc with centre S is drawn from T to R. A point P on this arc is 1 unit from TU and 8 units from RU. Suppose we wanted to find the the length of the side of square RSTU.Let the length of the square be x.Draw a line in your diagram horizontally right from the point P to form a right-angled triangle. Find the lengths of the three sides of your triangle in terms of x (you do not at this stage need to find the value of x).5156200317500[Kangaroo Pink 2015 Q9] In the grid, each small square has side of length 1. What is the minimum distance from ‘Start’ to ‘Finish’ travelling only on edges or diagonals of the squares?A 22B 10+2C 2+22D 42E 6 [IMC 2011 Q19] Harrogate is 23km due north of Leeds, York is 30km due east of Harrogate, Doncaster is 48km due south of York, and Manchester is 70km due west of Doncaster. To the nearest kilometre, how far is it from Leeds to Manchester, as the crow flies?A 38kmB 47kmC 56kmD 65kmE 74km[IMC 2009 Q20] A square, of side two units, is folded in half to form a triangle. A second fold is made, parallel to the first, so that the apex of this triangle folds onto a point on its base, thereby forming an isosceles trapezium. What is the perimeter of this trapezium?A 4+2B 4+22C 3+22D 2+32E 55135880698500[Kangaroo Pink 2010 Q7] The diagram shows a square PQRS and two equilateral triangles RSU and PST. PQ has length 1. What is the length of TU?A 2B 32C 3D 5-1E 6-1 Find the area of a regular octagon of side length 1.49530006985[Based on Hamilton 2006 Q4] A circle is inscribed in a square and a rectangle is placed inside the square but outside the circle. Two sides of the rectangle lie along sides of the square and one vertex lies on the circle, as shown. The rectangle is twice as high as it is wide. We wish to find the ratio of the area of the square to the area of the rectangle.a) Using the information, give suitable numeric lengths to the width and height of the rectangle.b) As per the usual advice for diagrams involving circles, add a suitable line to your diagram, and form a right-angle triangle. If the radius of your circle is r, give expressions for the three lengths of your triangle.49244251143000 [IMC 2012 Q21] The parallelogram PQRS is formed by joining together four equilateral triangles of side 1 unit, as shown. What is the length of the diagonal SQ?A 7B 8C 3D 6E 552444651397000[IMC 2004 Q18] In the triangle PQR, there is a right angle at Q and angle QPR is 60°. The bisector of the angle QPR meets QR at S, as shown. What is the ratio QS:SR?A 1:1B 1:2C 1:3-3D 1:3E 1:2[Cayley 2014 Q4] The square ABIJ lies inside the regular octagon ABCDEFGH. The sides of the octagon have length 1. Find length CJ.[JMO Mentoring May2012 Q4] A triangle has two angles which measure 30° and 105°. The side between these angles has length 2 cm. What is the perimeter of the triangle?5448935381000[IMC 2004 Q23] In the diagram, the letter S is made from two arcs, KL and MN, which are each five-eighths of the circumference of a circle of radius 1, and the line segment LM, which is tangent to both circles. At points K and N, common tangents to the two circles touch one of the circles. What is the length LM?A 32B 3-2C 2D 322E 1+247955201397000[IMC 1999 Q25] A rectangular sheet of paper with sides 1 and 2 has been folded once as shown, so that one corner just meets the opposite long edge. What is the value of the length d?A 12B 2-1C 716D 3-2E 235200650952500[Kangaroo Pink 2008 Q24] In the diagram, KLMN is a unit square. Arcs of radius one unit are drawn using each of the four corners of the squares as centres. The arcs centred at K and L intersect at Q; the arcs centred at M and N intersect at P. What is the length of PQ?A 2-2B 34C 5-2D 33E 3-1Exercise 2 – Using Variables and Pythagoras5214876633400[SMC 2014 Q19] The diagram shows a quadrant of radius 2, and two touching semicircles. The larger semicircle has radius 1. What is the radius of the smaller semicircle?A π6B 32C 12D 13E 23506285563500[IMC 2015 Q24] In square RSTU a quarter-circle arc with centre S is drawn from T to R. A point P on this arc is 1 unit from TU and 8 units from RU. What is the length of the side of square RSTU?A 9B 10C 11D 12E 1352482752667000[IMC 2011 Q23] A window frame in Salt’s Mill consists of two equal semicircles and a circle inside a large semicircle with each touching the other three as shown. The width of the frame is 4m. What is the radius of the circle in metres?A 23B 22C 34D 22-1E 152628805143500[IMC 2010 Q23] The diagram shows a pattern of eight equal shaded squares inside a circle of area π square units. What is the area (in square units) of the shaded region?A 113B 135C 123D 179E 2 51892201206500[Cayley 2010 Q5] A square sheet of paper ABCD is folded along FG, as shown, so that the corner B is folded onto the midpoint M of CD. Prove that the sides of triangle GCM have lengths in the ratio 3:4:5.33961201550400[Cayley 2008 Q5] A kite has sides AB and AD of length 25cm and sides CB and CD of length 39cm. The perpendicular distance from B to AD is 24cm. The perpendicular distance from B to CD is h cm.Find the value of h.[Hamilton 2010 Q5] The diagram shows three touching circles, whose radii are a, b and c, and whose centres are at the vertices Q, R and S of a rectangle QRST. The fourth vertex T of the rectangle lies on the circle with centre S. Find the ratio a:b:c.54006751333500[Kangaroo Pink 2004 Q24] The shaded area is equal to 2π. What is the length of PQ?A 1 B 2C 3 D 4E impossible to determine49083031206500[Maclaurin 2011 Q5] Three circles touch the same straight line and touch each other, as shown. Prove that the radii a, b, and c, where c is smallest, satisfy the equation:1a+1b=1cExercise 3 – Areas and Circles50196751079500[JMO 2015 A5] Two circles of radius 1 cm fit exactly between two parallel lines, as shown in the diagram. The centres of the circles are 3 cm apart. What is the area of the shaded region bounded by the circles and the lines?5400675698500[Kangaroo Grey 2005 Q9] In the diagram, the five circles have the same radii and touch as shown. The square joins the centres of the four outer circles. The ratio of the area of the shaded parts of all five circles to the area of the unshaded parts of all five circles isA 5:4B 2:3C 2:5D 1:4E 1:35272405317500[IMC 2014 Q20] The diagram shows a regular pentagon and five circular arcs. The sides of the pentagon have length 4. The centre of each arc is a vertex of the pentagon, and the ends of the arc are the midpoints of the two adjacent edges. What is the total shaded area?A 8πB 10πC 12πD 14πE 16π[IMC 2014 Q10] The diagram shows five touching semicircles, each with radius 2.What is the length of the perimeter of the shaded shape?A 5πB 6πC 7πD 8πE 9π [Hamilton 2011 Q4] A square just fits within a circle, which itself just fits within another square, as shown in the diagram. Find the ratio of the two shaded areas.5039360000[IMC 1997 Q23] The square ABCD is inscribed in a circle of radius 1. Semicircles are drawn with diameters AB, BC, CD, DA as shown, and the parts of these semicircles which lie outside the original circle are shaded. What is the total area of these four shaded regions?A 3π+2B 2C πD 1E π24921073968700[Cayley 2009 Q2] The boundary of a shaded figure consists of four semicircular arcs whose radii are all different. The centre of each arc lies on the line AB, which is 10cm long.What is the length of the perimeter of the figure.5536551571500[IMC 2008 Q17] The shaded region is bounded by eight equal circles with centres at the corners and midpoints of the sides of a square. The perimeter of the square has length 8. What is the length of the perimeter of the shaded region?A πB 2πC 8D 3πE 4π 54578251460500[Kangaroo Pink 2007 Q18] A coin with diameter 1 cm rolls around the outside of a regular hexagon with edges of length 1cm until it returns to its original position. In centimetres, what is the length of the path traced out by the centre of the coin?A 6+π2B 12+πC 6+πD 12+2πE 6+2π5105400571500[IMC 2006 Q21] The diagram shows two semicircular arcs, PQRS and QOR. The diameters, PS and QR, of the two semicircles are parallel; PS is of length 4 and is a tangent to semicircular arc QOR. What is the area of the shaded region?A 2π-2B 3π C πD 4E 2π-45124450317500[STMC Regional 2007/08 Q5] The square in the diagram below has sides of length two units. The shaded sections are enclosed by 4 semi-circles. Calculate the exact value of the total area of the unshaded regions.48094901333500[Hamilton 2005 Q2] The region shown shaded in the diagram is bounded by three touching circles of radius 1 and the tangent to two of the circles. Calculate the perimeter of the shaded region.50895251460500[Hamilton 2010 Q4] The diagram shows a quarter-circle with centre O and two semicircular arcs with diameters OA and OB. Calculate the ratio of the area of the region shaded grey to the area of the region shaded black.5019675635000[Kangaroo Grey 2005 Q18] Two rectangles ABCD and DBEF are shown in the diagram. What is the area of the rectangle DBEF?A 10cm2B 12cm2C 13cm2D 14cm2E 16cm242767254572000[IMC 2003 Q19] What is the area of the pentagon shown?A 12ab-cB 12ba+cC 12a(b+c)D 12bc-aE 12ca+b5326380-508000[IMC 2005 Q18] Three-quarters of the area of the rectangle has been shaded. What is the value of x?A 2B 2.4C 3D 3.6E 452717411841500[Cayley 2015 Q5] The diagram shows a right-angled triangle and three circles. Each side of the triangle is a diameter of one of the circles. The shaded region R is the region inside the two smaller circles but outside the largest circle. Prove that the area of R is equal to the area of the triangle.53054251206500[Kangaroo Grey 2012 Q17] In the diagram, WXYZ is a square, M is the midpoint of WZ and MN is perpendicular to WY. What is the ratio of the area of the shaded triangle MNY to the area of the square?A 1:6B 1:5C 7:36D 3:16E 7:405106035000[IMC 1998 Q24] Each of the sides of this regular octagon has length 2cm. What is the difference between the area of the shaded region and the area of the unshaded region (in cm2)?A 0B 1C 1.5D 2E 22381028575000Exercise 4: Cutting and Reforming Test Your Understanding: The figure shows two shapes that fit together exactly. Each shape is formed by four semicircles of radius 1. What is the total shaded area?5046345139065Test Your Understanding: The diagram shows a design formed by drawing six lines in a regular hexagon. The lines divide each edge of the hexagon into three equal parts. What fraction of the hexagon is shaded?A 15 B 29 C 14 D 310 E 51653037849692Question 1: The diagram contains six equilateral triangles with sides of length 2 and a regular hexagon with sides of length 1.What fraction of the whole shape is shaded?A 18 B 17 C 16 D 15 E 1451701706140450053955956985Question 2: The diagram shows an equilateral triangle with its corners at the mid-points of alternate sides of a regular hexagon. What fraction of the area of the hexagon is shaded?A 12 B 13 C 38 D 49 E 712Question 3: Find the fraction of the shape which is shaded.5121275825500Question 4: The diagram on the right shows a rectangle with sides of length 5cm and 4cm. All the arcs are quarter-circles of radius 2cm. What is the total shaded area in cm2.53698967895Question 5: The diagram shows a ceramic design by the Catalan architect Antoni Gaudi. It is formed by drawing eight lines connecting points which divide the edges of the outer regular octagon into three equal parts, as shown.What fraction of the octagon is shaded?A 1/5B 2/9C 1/4D 3/10E 5/1653120509621Question 6: The diagram shows a square with two lines from a corner to the middle of an opposite side. The rectangle fits exactly inside these two lines and the square itself. What fraction of the square is occupied by the shaded rectangle?A 13 B 25 C 310 D 12 E 38 ................
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