A right-angled triangle has two shorter sides of length 6 cm and 8 cm. Diagram: right-angled triangle with the two shorter sides marked 6 cm and 8 cm meeting at the right angle, hypotenuse unmarked labelled x cm. Calculate the length of the hypotenuse.
Diagram NOT accurately drawn
(Total for Question 1 is 2 marks)
2
A right-angled triangle has a hypotenuse of length 13 cm and one shorter side of length 5 cm. Diagram: right-angled triangle, hypotenuse marked 13 cm, one shorter side marked 5 cm, the other shorter side unmarked labelled y cm. Work out the length of the third side.
Diagram NOT accurately drawn
(Total for Question 2 is 2 marks)
3
A right-angled triangle has two shorter sides of length 5.2 cm and 7.9 cm. Diagram: right-angled triangle with the two shorter sides marked 5.2 cm and 7.9 cm meeting at the right angle, hypotenuse unmarked labelled h cm. Calculate the length of the hypotenuse. Give your answer correct to 1 decimal place.
Diagram NOT accurately drawn
(Total for Question 3 is 3 marks)
4
A rectangular gate measures 12 m wide by 5 m tall, with a diagonal metal brace fitted from one corner to the other. Diagram: rectangle representing a gate, width marked 12 m along the base, height marked 5 m up the side, a diagonal brace drawn from bottom-left to top-right labelled d m. Calculate the length of the diagonal brace, d.
Diagram NOT accurately drawn
(Total for Question 4 is 2 marks)
5
A triangle has sides of length 9 cm, 12 cm and 15 cm. Determine, showing your working, whether this triangle is right-angled.
(Total for Question 5 is 3 marks)
6
An isosceles triangle has two equal sides of length 13 cm and a base of length 10 cm. A line is drawn from the apex perpendicular to the base, meeting the base at its midpoint. Diagram: isosceles triangle with the two equal sides marked 13 cm, base marked 10 cm, and a dashed perpendicular line from the apex to the midpoint of the base labelled h cm.
Diagram NOT accurately drawn
(a)State the lengths of the two parts the base is split into by the perpendicular line, and explain why they are equal.(1)
(b)Calculate the perpendicular height, h, of the triangle.(3)
(Total for Question 6 is 4 marks)
7
A ladder of length 6.5 m leans against a vertical wall. The foot of the ladder is 2.5 m from the base of the wall on horizontal ground. Diagram: right-angled triangle formed by the wall, the ground and the ladder; the ladder (hypotenuse) marked 6.5 m, the distance of the foot of the ladder from the wall marked 2.5 m, and the height up the wall labelled h m. Calculate how far up the wall the ladder reaches.
Diagram NOT accurately drawn
(Total for Question 7 is 3 marks)
8
Point A has coordinates (1, 2) and point B has coordinates (4, 6). Work out the distance AB.
(Total for Question 8 is 3 marks)
9
A ship sails 15 km due east from port P to point Q, then sails 8 km due south from Q to point R. Diagram: port P at the start, a horizontal arrow of length 15 km pointing east to point Q, then a vertical arrow of length 8 km pointing south to point R, and a dashed straight line drawn directly from P to R. Calculate the direct distance from P to R.
Diagram NOT accurately drawn
(Total for Question 9 is 3 marks)
10
Fatima is building a garden shed. The cross-section of the shed is a rectangle 8 m wide and 2.5 m tall, with a triangular roof on top. The roof has a horizontal span of 8 m (the same as the width of the rectangle) and rises to a ridge 3 m above the top of the rectangle, positioned centrally. Diagram: rectangle 8 m wide and 2.5 m tall, with an isosceles triangle roof on top; horizontal span 8 m and vertical rise 3 m to the central ridge, shown with a dashed line.
Diagram NOT accurately drawn
(a)Calculate the length of one sloping roof edge.(3)
(b)Fatima needs to buy timber to edge the two sloping roof edges and the two vertical walls of the cross-section (the base is not edged). Calculate the total length of timber needed.(2)
(Total for Question 10 is 5 marks)
11
An isosceles trapezium has parallel sides of length 14 cm and 8 cm, and slanting (non-parallel) sides of length 5 cm each. Diagram: isosceles trapezium, longer parallel side marked 14 cm at the base, shorter parallel side marked 8 cm at the top, both slanting sides marked 5 cm, with a dashed perpendicular line from one end of the top side down to the base labelled h cm. Calculate the height of the trapezium, h.
Diagram NOT accurately drawn
(Total for Question 11 is 4 marks)
12
A rhombus has diagonals of length 16 cm and 12 cm. The diagonals of a rhombus bisect each other at right angles. Diagram: rhombus with its two diagonals drawn, one diagonal marked 16 cm and the other marked 12 cm, crossing at right angles at the centre. Calculate the length of one side of the rhombus.
Diagram NOT accurately drawn
(Total for Question 12 is 3 marks)
13
A cone has base radius 5 cm and perpendicular height 12 cm. Diagram: cone shown in cross-section, base radius marked 5 cm from the centre of the base to the edge, perpendicular height marked 12 cm from the centre of the base to the apex, and the slant height labelled l cm from the apex to the edge of the base. Calculate the slant height, l, of the cone.
Diagram NOT accurately drawn
(Total for Question 13 is 3 marks)
14
An isosceles triangle has two equal sides of length 17 cm and a base of length 16 cm. Diagram: isosceles triangle with equal sides marked 17 cm, base marked 16 cm, and a dashed perpendicular height from the apex to the midpoint of the base labelled h cm.
Diagram NOT accurately drawn
(a)Calculate the perpendicular height, h, of the triangle.(3)
(b)Hence calculate the area of the triangle.(2)
(Total for Question 14 is 5 marks)
15
A circle has centre O and radius 6 cm. PT is a tangent to the circle at point T, so OT is perpendicular to PT. The distance from the external point P to the centre O is 10 cm. Diagram: circle with centre O, radius OT marked 6 cm, tangent line PT drawn from an external point P touching the circle at T with a right angle marked between OT and PT, and the line OP marked 10 cm. Calculate the length of the tangent, PT.
Diagram NOT accurately drawn
(Total for Question 15 is 3 marks)
16
A kite PQRS has PR as its line of symmetry. The diagonals PR and QS cross at right angles at point M, where PM = 8 cm, MR = 15 cm, and QM = MS = 6 cm. Diagram: kite PQRS with diagonal PR drawn as the line of symmetry and diagonal QS crossing it at right angles at point M; PM marked 8 cm, MR marked 15 cm, QM and MS each marked 6 cm.
Diagram NOT accurately drawn
(a)Calculate the length of PQ.(2)
(b)Calculate the length of QR, giving your answer correct to 1 decimal place.(3)
(Total for Question 16 is 5 marks)
17
A cuboid has length 12 cm, width 5 cm and height 4 cm. Diagram: cuboid with length 12 cm, width 5 cm and height 4 cm, a diagonal drawn across the base labelled b cm, and a space diagonal drawn from one bottom corner to the diagonally opposite top corner labelled D cm.
Diagram NOT accurately drawn
(a)Calculate b, the length of the diagonal of the base.(2)
(b)Calculate the length of the space diagonal, D, giving your answer correct to 1 decimal place.(3)
(Total for Question 17 is 5 marks)
18
Callum is an electrician who needs to run a cable in a straight line between the tops of two vertical poles standing on horizontal ground. The poles are 24 m apart. One pole is 10 m tall and the other is 15 m tall. Diagram: two vertical poles on horizontal ground, 24 m apart, one pole 10 m tall and the other 15 m tall, a straight cable drawn from the top of one pole to the top of the other, and a dashed line showing the height difference between the pole tops. Calculate the length of cable needed, giving your answer correct to 1 decimal place.
Diagram NOT accurately drawn
(Total for Question 18 is 4 marks)
19
A right-angled triangle has legs of length x cm and (x + 7) cm, and hypotenuse (x + 8) cm. Diagram: right-angled triangle with the two shorter sides labelled x cm and (x + 7) cm meeting at the right angle, and the hypotenuse labelled (x + 8) cm.
Diagram NOT accurately drawn
(a)Form an equation in x and show that it simplifies to x2 - 2x - 15 = 0.(2)
(b)Solve the equation to find the value of x, explaining why the other solution is rejected.(2)
(c)State the lengths of the three sides of the triangle.(1)
(Total for Question 19 is 5 marks)
20
A wooden storage box has internal dimensions 30 cm by 40 cm by 22.5 cm. Priti wants to know whether a thin metal rod of length 55 cm will fit inside the box in a single straight line. Diagram: cuboid box with internal dimensions 40 cm, 30 cm and 22.5 cm, with the space diagonal drawn from one bottom corner to the diagonally opposite top corner, labelled D cm. Determine, showing your working, whether the rod fits inside the box.
Diagram NOT accurately drawn
(Total for Question 20 is 5 marks)
Mark scheme · 4.9D Pythagoras: Fluency and Exam Drill
Question 1
M1 62 + 82 oe (= 100)
A1 10 cm cao
Answer: 10 cm
Question 2
M1 132 - 52 oe (= 144)
A1 12 cm cao
Answer: 12 cm
Question 3
M1 5.22 + 7.92 oe (= 89.45)
M1√their 89.45
A1 awrt 9.5 cm
Answer: 9.5 cm (1 d.p.)
Question 4
M1 122 + 52 oe (= 169)
A1 13 m cao
Answer: 13 m
Question 5
M1 92 + 122 (= 225)
M1 152 (= 225)
C1 correct conclusion: since 92 + 122 = 152 (225 = 225), the triangle is right-angled cso
(a) B1 5 cm and 5 cm, because the perpendicular from the apex of an isosceles triangle bisects the base by symmetry oe
(a) Answer: 5 cm and 5 cm
(b) M1 132 - 52 oe (= 144)
(b) M1√their 144
(b) A1 12 cm cao
(b) Answer: 12 cm
Question 7
M1 6.52 - 2.52 oe (= 36)
M1√their 36
A1 6 m cao
Answer: 6 m
Question 8
M1 (4-1)2 + (6-2)2 oe (= 25)
M1√their 25
A1 5 cao
Answer: 5
Question 9
M1 152 + 82 oe (= 289)
M1√their 289
A1 17 km cao
Answer: 17 km
Question 10
(a) M1 42 + 32 oe (= 25), using half the span = 8 / 2 = 4
(a) M1√their 25
(a) A1 5 m cao
(a) Answer: 5 m
(b) M1 2 x their 5 + 2 x 2.5 oe
(b) A1 15 m cao
(b) Answer: 15 m
Question 11
M1 (14 - 8) / 2 oe (= 3)
M1 52 - 32 oe (= 16)
M1√their 16
A1 4 cm cao
Answer: 4 cm
Question 12
M1√82 + 62 oe
M1√100
A1 10 cm cao
Answer: 10 cm
Question 13
M1 recognises right-angled triangle formed by the radius, height and slant height oe
M1 52 + 122 oe (= 169)
A1 13 cm cao
Answer: 13 cm
Question 14
(a) M1 172 - 82 oe (= 225), using half the base = 16 / 2 = 8
(a) M1√their 225
(a) A1 15 cm cao
(a) Answer: 15 cm
(b) M1 0.5 x 16 x their 15 oe
(b) A1 120 cm2 cao
(b) Answer: 120 cm2
Question 15
M1 102 - 62 oe (= 64)
M1√their 64
A1 8 cm cao
Answer: 8 cm
Question 16
(a) M1√82 + 62 oe
(a) A1 10 cm cao
(a) Answer: 10 cm
(b) M1 152 + 62 oe (= 261)
(b) M1√their 261
(b) A1 awrt 16.2 cm
(b) Answer: 16.2 cm (1 d.p.)
Question 17
(a) M1 122 + 52 oe (= 169)
(a) A1 13 cm cao
(a) Answer: 13 cm
(b) M1 their b2 + 42 oe, i.e. 169 + 16 (= 185)
(b) M1√their 185
(b) A1 awrt 13.6 cm
(b) Answer: 13.6 cm (1 d.p.)
Question 18
M1 15 - 10 oe (= 5), the height difference between the poles
M1 242 + 52 oe (= 601)
M1√their 601
A1 awrt 24.5 m
Answer: 24.5 m (1 d.p.)
Question 19
(a) M1 x2 + (x + 7)2 = (x + 8)2 oe
(a) C1 correct expansion and simplification to x2 - 2x - 15 = 0 cso
(a) Answer: x2 - 2x - 15 = 0 (shown)
(b) M1 (x - 5)(x + 3) = 0 oe, or correct use of the quadratic formula
(b) A1 x = 5, rejecting x = -3 since a length cannot be negative
(b) Answer: x = 5 (x = -3 is rejected as a length cannot be negative)
(c) B1 5 cm, 12 cm, 13 cm cao (ft their x)
(c) Answer: 5 cm, 12 cm, 13 cm
Question 20
M1 302 + 402 oe (= 2500)
M1 their 2500 + 22.52 oe (= 3006.25)
M1√their 3006.25
A1 awrt 54.8 cm
B1 correct conclusion with valid comparison to 55 cm, e.g. since 54.8 < 55, the rod does NOT fit, because the space diagonal is the longest straight line that fits inside the box (ft, must not round up to conclude incorrectly that it fits)
Answer: No, the rod does not fit, since the internal space diagonal is approximately 54.8 cm, which is less than the 55 cm rod.