Year 9 Maths • AC9M9M03 • Authored homework
Pythagoras, trigonometry and similarity
Solve spatial problems using right triangles, Pythagoras, simple trigonometric ratios, scale and similarity. Show diagrams, units and checks for reasonableness. Show working, use correct units or notation, and check reasonableness.
Part A
Short-answer questions
Find the hypotenuse of a right triangle with shorter sides 9 cm and 12 cm.
A right triangle has hypotenuse 13 m and one shorter side 5 m. Find the other shorter side.
State the sine, cosine and tangent ratios using opposite, adjacent and hypotenuse.
In a right triangle, identify the hypotenuse and explain how you know.
A ladder is 4 m from a wall and reaches 7 m up the wall. Estimate the ladder length.
Find the scale factor from a triangle with side 6 cm to a similar triangle with matching side 15 cm.
Explain why similar triangles have equal matching angles.
Use tan θ = opposite/adjacent to write an equation for an unknown angle.
Correct this error: a^2 + b^2 = c can be used for Pythagoras.
Give one practical situation where right-triangle measurement is useful.
Part B
Long-answer questions
A 6 m ladder leans against a wall with its base 2 m from the wall. Find the height it reaches and explain your method.
A ramp rises 1.2 m over a horizontal distance of 4.5 m. Find the ramp length and estimate its angle to the ground.
Use similarity to find the height of a tree if a 1.5 m student casts a 2 m shadow and the tree casts a 10 m shadow.
Compare solving a right-triangle problem using Pythagoras and using trigonometry. State when each method is suitable.
A scale drawing uses 1 cm to represent 2.5 m. A path is 7.2 cm on the drawing. Find its real length.
A student uses the longest given side as the hypotenuse without checking the right angle. Explain the risk and correct approach.
Solve a navigation-style problem involving north/east distances and straight-line distance. Include a diagram.
Create a similar-triangles problem with missing side length, solve it and check with scale factor.
Explain how angle properties, scale and right-triangle methods can work together in a spatial problem.
Write a full worked solution to a measurement problem involving a right triangle, including diagram, formula, substitution, answer and units.
Part C
Research and understanding task
Research one real application of Pythagoras, trigonometry or similarity, such as building ramps, surveying, navigation, shadows or phone map distance. Create and solve a related measurement problem.