Year 8 Mathematics | Victorian Curriculum 2.0
Pythagoras' theorem
Topic 13 | Measurement & Space | Answer key

Year 8 core - answers

Fluency

Find the hypotenuse

    1. 55. Method: 9+16=259 + 16 = 25; 25=5\sqrt{25} = 5.
    2. 1313. Method: 25+144=16925 + 144 = 169.
    3. 1010. Method: 36+64=10036 + 64 = 100.
    4. 4141. Method: 81+1600=1681=41281 + 1600 = 1681 = 41^2.
    5. 8=22≈2.83\sqrt{8} = 2\sqrt{2} \approx 2.83. Method: 4+4=84 + 4 = 8.
    6. 22. Method: 1+3=41 + 3 = 4; 4=2\sqrt{4} = 2.
Fluency

Find a leg

    1. 2424. Method: 625−49=576625 - 49 = 576.
    2. 1515.
    3. 1212.
    4. 4040. Method: 2500−900=16002500 - 900 = 1600.
    5. 44. Method: 20−4=1620 - 4 = 16.
Fluency

Is it a right triangle?

    1. Yes. 36+64=10036 + 64 = 100.
    2. No. 16+25=41≠3616 + 25 = 41 \ne 36.
    3. Yes. 49+576=62549 + 576 = 625.
    4. No. 25+144=169≠19625 + 144 = 169 \ne 196.
Reasoning

Explain and spot the mistake

    1. Lee took the square root of the sum incorrectly. 52+122=25+144=1695^2 + 12^2 = 25 + 144 = 169, and 169=13\sqrt{169} = 13, not 1717.
    2. The hypotenuse satisfies c2=a2+b2c^2 = a^2 + b^2. If either leg had length ≥c\geq c, the equation would fail. Geometrically, the hypotenuse is opposite the largest angle (90∘90^\circ), which corresponds to the longest side.
    3. Yes: 81+144=22581 + 144 = 225. It is similar to (3,4,5)(3, 4, 5) with scale factor 33.
    4. No. 72+242=49+576=625=2527^2 + 24^2 = 49 + 576 = 625 = 25^2, so the triple is (7,24,25)(7, 24, 25). (7,24,26)(7, 24, 26) has 625≠676625 \ne 676, so it is not a right triangle.
Problem solving

Real contexts

    1. 32≈5.66\sqrt{32} \approx 5.66 m.
    2. 5050 m.
    3. 1717 m. (8,15,178, 15, 17 triple.)
    4. Yes, ≈21.8\approx 21.8 inches. 502−452=475≈21.79\sqrt{50^2 - 45^2} = \sqrt{475} \approx 21.79.
    5. 1010 km. (6,8,106, 8, 10.)

Challenge - answers

Reasoning

Harder triangles

    1. Legs aa and aa: c2=2a2c^2 = 2a^2; c=a2c = a\sqrt{2}.
    2. 1313 cm. Method: base diagonal 32+42=5\sqrt{3^2 + 4^2} = 5; body diagonal 52+122=13\sqrt{5^2 + 12^2} = 13.
    3. 1313 cm. Method: half-diagonals 55 and 1212; 52+122\sqrt{5^2 + 12^2}.
    4. 8.78.7 cm. Method: height =102−52=75≈8.66= \sqrt{10^2 - 5^2} = \sqrt{75} \approx 8.66.
Year 8 Mathematics study companion | Answer key