Geometry and measures worksheet — GCSE Higher

1.Triangle ABC is similar to triangle PQR. AB corresponds to PQ, and BC corresponds to QR. AB = 5 cm, PQ = 15 cm and BC = 7 cm. Work out the length of QR.

2.A circular garden pond has a diameter of 5 m. A gardener is laying gravel around it in a ring-shaped path 1 m wide, measured outward from the edge of the pond. Work out the area of the gravel path. Use π = 3.14.

3.OABC is a parallelogram, with OA = a and OC = c. M is the midpoint of AB. Express the vector MC in terms of a and c.

4.A triangular plot of land has sides AB = 13 m and AC = 10 m, with angle BAC = 72° between them. Work out the perimeter of the plot. Give your answer to 1 decimal place.

5.Each of these has an exact value. Write down the one whose value is the greatest.

6.A scale drawing shows a room that is 5 cm wide. The real room is 15 m wide. Write the scale of the drawing in the form 1 : n.

7.A game designer places a coin at (5, 3) on a grid. The game rotates the coin 90° clockwise about the point (2, 3) each time the player presses a button, and this is applied twice in a row. Work out the coordinates of the coin after the button is pressed twice.

8.Triangle ABC is drawn. The interior angle bisectors from vertices A and B are constructed and meet at point I inside the triangle. Which statement about point I must be true?

9.The sum of the interior angles of a polygon is 1980°. Work out the number of sides of the polygon.

10.In triangle OAB, OA = a and OB = b. P is the point on OA such that OP = (1/3)a, and Q is the point on OB such that OQ = (1/3)b. Express the vector PQ in terms of a and b.

11.ABCD is a convex quadrilateral field with diagonal AC = 20 m. In triangle ABC, AB = 14 m and angle BAC = 35°. In triangle ACD, AD = 16 m and angle DAC = 40°. Work out the total area of the field. Give your answer to 1 decimal place.

12.The midpoint of the line segment AB is (3, 5). A is the point (1, 3). Work out the coordinates of B.

13.An ice cream is made from a cone of radius 3 cm and height 10 cm, topped with a hemisphere of the same radius sitting exactly on top of the cone. Work out the total volume of the ice cream. Use π = 3.14. Give your answer to the nearest whole number. Volume of a cone = 1/3 × πr²h. Volume of a sphere = 4/3 × πr³.

14.An isosceles triangle ABC has AB equal to AC. AM is a line from vertex A, perpendicular to BC, meeting BC at point M. Which condition proves that triangle ABM is congruent to triangle ACM?

15.Triangle XYZ has angle X = 90°. Angle Y = 34°. Work out angle Z.