Stage 9 - Q1 - 2 marks
A rectangular courtyard is 24 m by 45 m. Compare the shortest route between opposite corners with walking along two edges. How much distance is saved?
Hint
Compare two routes joining the same endpoints.
Stage 9 - Q2 - 2 marks
A triangle has side lengths 32, 60 and 68 cm. Prove that it is right-angled.
Hint
Test the longest side as the possible hypotenuse.
Stage 9 - Q3 - 3 marks
A triangle has sides 14, 48 and 51 cm. Is it right-angled? Justify exactly.
Hint
An almost-right diagram is not an exact right angle.
Stage 9 - Q4 - 3 marks
A square has side 14 cm. Find its diagonal in simplest exact root form.
Hint
Factor the square number out of the root.
Stage 9 - Q5 - 4 marks
\(AB=5\) cm, \(BC=8\) cm and \(CD=2\) cm. The marked angles \(ABC\) and \(ACD\) are right angles. Find \(AD\) exactly.
Hint
Avoid rounding the shared irrational length.
Stage 9 - Q6 - 4 marks
\(AB=7\) cm, \(BC=9\) cm and \(CD=2\) cm. The marked angles \(ABC\) and \(ACD\) are right angles. Find \(AD\) exactly.
Hint
Avoid rounding the shared irrational length.
Stage 9 - Q7 - 5 marks
A square of side 20 cm and a rectangle with sides 15 cm and 25 cm have equal perimeters. Compare their diagonals exactly.
Hint
You do not need decimal approximations to compare positive roots.
Stage 9 - Q8 - 5 marks
A square has diagonal 10 cm. Find its side in exact root form and its area exactly.
Hint
Solve for the side square before taking a root.
Stage 9 - Q9 - 5 marks
One side of a right triangle is 16 cm. The other two sides are positive integers no greater than 40 cm. Find every possible unordered pair of other lengths.
Hint
For each possible hypotenuse up to 40, subtract the known leg square; also test the known side as hypotenuse.
Stage 9 - Q10 - 3 marks
A rectangle has perimeter 40 cm and diagonal \(\sqrt{208}\) cm. Find its two side lengths exactly and show that they are determined up to swapping.
Hint
Equal deviations from the average side length remove the linear terms.