Triangle ABC is drawn on a 1cm2 grid.
E is the point (0, 0).
(a) Write down the gradient of the line AB.
(b) The gradient of BC is -0.5.
Write down the equation of the line BC in the form y = mx + c.
(c) Write down the ratio AE:EC.
Give your answer in its simplest form.
(d) Measure angle ABE.
(e) Triangle ABE is similar to triangle BCE.
Explain what the word similar tells you about the triangles ABE and BCE.
(f) Calculate the area of triangle ABC.
(g) ABCD is a rectangle.
(i) Mark point D on the grid.
(ii) Write down the co-ordinates of D.
▶️ Answer/Explanation
(a) Ans: 2
Gradient = rise/run = (6-0)/(3-0) = 2
(b) Ans: y = -0.5x + 6
Using point B(6,3): 3 = -0.5(6) + c ⇒ c = 6
(c) Ans: 1:4
AE = 3 units, EC = 12 units ⇒ AE:EC = 3:12 = 1:4
(d) Ans: 26.6° (or 25°-29° accepted)
Using tanθ = opposite/adjacent = 1/2 ⇒ θ ≈ 26.6°
(e) Explanation:
Similar triangles have:
• Corresponding angles equal
(a)
(i) Explain why these rectangles are mathematically similar.
(ii) How many times bigger is the area of the large rectangle than the area of the small rectangle?
(b)
The diagram shows a net of a cube.
(i) The square labelled B is the base. Write the letter T in the square that is the top of the cube.
(ii) On the grid, draw a different net of this cube.
(c) The diagram shows a cuboid.
(i) Work out the surface area of this cuboid.
(ii) Work out the volume of this cuboid.
(iii) Write down the dimensions of a different cuboid that has the same volume as this cuboid.
▶️ Answer/Explanation
(a)(i) Ans: The rectangles are mathematically similar because one is an exact enlargement of the other with corresponding angles equal and sides in proportion (3:1 ratio).
(a)(ii) Ans: 9 times bigger
Area scale factor = (Linear scale factor)² = (3)² = 9
Small rectangle area = 2×3 = 6 cm²
Large rectangle area = 6×9 = 54 cm²
54 ÷ 6 =