51 a C2 2 C1 for explaining all sides are the same length H Joining adjacent midpoints forms a C1 for explaining how all angles are 90° right-angled isosceles triangle, with the perpendicular sides each half the length of the side of the square. All four triangles are congruent. At each midpoint, there are two angles of 45°, leaving the vertex of the new shape C1 C1 for communicating a given original side length and being 90°. Since all the sides are equal finding the length of a side of the inscribed square and all the angles are 90°< the new shape is a square. b Area of original rectangle = x2 M1 M1 for correctly using theorem to find the Pythagoras’ If the length of the side of the inscribed length of the side of the new square square is then: h, = h2 A1 A1 for correct expression of h2 = h2 = h2 The area of the inscribed square is half the area of the original square. Area of inscribed square is h2 A1 A1 for correctly stating area = C1 C1 for clear communication of complete solution 7 52 Each internal angle of an octagon is C2 2 C1 for explanation of 135° H 135°. Each internal angle of a hexagon is C1 for explanation of 120° 120°. The sum of these two angles is 255°. B1 B1 for 255° The sum of the angles in a quadrilateral is 360° so the sum of the remaining angles is: 360° 255° M1 for subtraction from 360° – M1 = 105°. A1 for 105° A1 The two remaining angles are equal, as C1 for explaining two angles are equal C2 the line joining through the vertices C, J, C1 for clear reasons given as to why G and L (and thus through the obtuse angles) is a line of symmetry. JFG = 52.5° B1 for correct 52.5° B1 8
51 a b Area of original rectangle = x2 If the length of the side of the inscribed square is h, then: h2 = h2 = h2 = Area of inscribed square is h2 = Joining adjacent midpoints forms a right-angled isosceles triangle, with the perpendicular sides each half the length of the side of the square. All four triangles are congruent. At each midpoint, there are two angles of 45°, leaving the vertex of the new shape being 90°. Since all the sides are equal and all the angles are 90°< the new shape is a square. The area of the inscribed square is half the area of the original square. C2 C1 M1 A1 A1 C1 2 C1 for explaining all sides are the same length C1 for explaining how all angles are 90° C1 for communicating a given original side length and finding the length of a side of the inscribed square M1 for correctly using Pythagoras’ theorem to find the length of the side of the new square A1 for correct expression of h2 A1 for correctly stating area C1 for clear communication of complete solution H 7 52 Each internal angle of an octagon is 135°. Each internal angle of a hexagon is 120°. The sum of these two angles is 255°. The sum of the angles in a quadrilateral is 360° so the sum of the remaining angles is: 360° – 255° = 105°. The two remaining angles are equal, as the line joining through the vertices C, J, G and L (and thus through the obtuse angles) is a line of symmetry. JFG = 52.5° C2 B1 M1 A1 C2 B1 2 C1 for explanation of 135° C1 for explanation of 120° B1 for 255° M1 for subtraction from 360° A1 for 105° C1 for explaining two angles are equal C1 for clear reasons given as to why B1 for correct 52.5° H 8
Higher_Skills_Geometry_MarkScheme_Edexcel.docx