Geometry and measures worksheet — GCSE Foundation

1.Triangle ABC has a right angle at B and hypotenuse AC = 17 cm, with AB = 8 cm. Triangle DEF has a right angle at E and hypotenuse DF = 17 cm. Which extra fact about DEF would complete the RHS condition for proving that ABC ≅ DEF, with the vertices matching in that order?

2.A drone flies from its base in three stages, each stage measured in metres east and north as a column vector. Stage 1 is the column vector with top number 30 and bottom number 40. Stage 2 is the column vector with top number −10 and bottom number 20. Stage 3 is the column vector with top number 15 and bottom number −5. Work out the column vector that would take the drone in a single straight flight back to its base from where it ends up.

3.p is the column vector with top number 4 and bottom number −3. Work out 2p, giving your answer as a column vector in the form (top, bottom).

4.Two of the angles in a triangle are 50° and 70°. Work out the size of the third angle.

5.A straight transversal crosses a pair of parallel lines. At one crossing point, the angle between the transversal and a parallel line is 118°. Work out the size of the co-interior (allied) angle at the other crossing point.

6.A triangular prism has a cross-section that is a triangle with a base of 3.5 cm and a perpendicular height of 4 cm. The prism is 10 cm long. Work out the volume of the prism.

7.In a right-angled triangle, the two shorter sides are a and b, and the hypotenuse is c. Write down the correct statement of Pythagoras' theorem.

8.Work out the exact value of sin 30° + cos 60°.

9.A roof has two triangular sections, P and Q. Section P has a sloping edge of 3.6 m and a base edge of 2.4 m, with an angle of 70° between them. Section Q has a sloping edge of 3.6 m and a base edge of 2.4 m, with an angle of 70° between them, arranged the same way round. A roofer wants to cut identical triangular felt panels for both sections without measuring Section Q separately. Which condition confirms that the two sections are congruent?

10.A solid cylinder is lying on its curved side on a table, with its circular ends facing left and right. What shape is its plan view, looking down from above?

11.Point E is at (4, 2). It is rotated 180° about the point (1, 1). Work out the coordinates of the image of point E.

12.Triangle T has a vertex at (2, 1). It is enlarged by a scale factor of 3, centre (0, 0). Work out the coordinates of the image of this vertex.

13.To construct the perpendicular from a point P to a line l, where P is a point above l, an arc centred at P is drawn to cross l at two points, X and Y. What is the correct next step?

14.Work out the exact value of (cos 45°)² + (sin 45°)².

15.A ramp's sloped surface crosses two horizontal parallel rails. At the top rail, the angle between the ramp and the rail on the right of the ramp is (3x + 10)°. At the bottom rail, the angle between the ramp and the rail on the left of the ramp is (5x − 30)°. Work out the value of x.