Topics › KS3 › Geometry & measures

Pythagoras

1. pyth-01 Present from here

   What is the length of the hypotenuse $x$? Give answers to 1 decimal place where needed.

   
   A Option A: $10\text{ cm}$

   B Option B: $14\text{ cm}$

   C Option C: $100\text{ cm}$

   D Option D: $5.3\text{ cm}$
   Answer and misconceptions

   A Option A: $10\text{ cm}$ ✓ Correct

   B Option B: $14\text{ cm}$ ✗

   Adds the side lengthsCalculates 6 + 8 instead of using the squares of the sides.

   C Option C: $100\text{ cm}$ ✗

   Forgets to square root at the endFinds 6² + 8² = 100 and stops; x² = 100, so x = 10.

   D Option D: $5.3\text{ cm}$ ✗

   Subtracts squares to find the hypotenuseCalculates √(8² − 6²); subtraction is only for finding a shorter side.
2. pyth-02 Present from here

   What is the length of side $x$? Give answers to 1 decimal place where needed.

   
   A Option A: $13.9\text{ cm}$

   B Option B: $8\text{ cm}$

   C Option C: $144\text{ cm}$

   D Option D: $12\text{ cm}$
   Answer and misconceptions

   A Option A: $13.9\text{ cm}$ ✗

   Always adds the squaresCalculates √(13² + 5²), treating x as the hypotenuse; x is a shorter side.

   B Option B: $8\text{ cm}$ ✗

   Subtracts the lengths, not their squaresCalculates 13 − 5 without squaring.

   C Option C: $144\text{ cm}$ ✗

   Forgets to square root at the endFinds 13² − 5² = 144 and stops; x² = 144, so x = 12.

   D Option D: $12\text{ cm}$ ✓ Correct
3. pyth-03 Present from here

   Which equation is true for this right-angled triangle?

   
   A Option A: $a^2 + b^2 = c^2$

   B Option B: $b^2 + c^2 = a^2$

   C Option C: $b + c = a$

   D Option D: $a^2 + c^2 = b^2$
   Answer and misconceptions

   A Option A: $a^2 + b^2 = c^2$ ✗

   Assumes c is always the hypotenuseUses the memorised a² + b² = c² without checking which side is opposite the right angle.

   B Option B: $b^2 + c^2 = a^2$ ✓ Correct

   C Option C: $b + c = a$ ✗

   Adds lengths without squaringThinks the two shorter sides add up to the hypotenuse.

   D Option D: $a^2 + c^2 = b^2$ ✗

   Takes the horizontal base as the hypotenuseThe hypotenuse is the side opposite the right angle, which is a.
4. pyth-04 Present from here

   What is the distance between the points $(1, 2)$ and $(4, 6)$? Give answers to 1 decimal place where needed.

   A Option A: $7$

   B Option B: $25$

   C Option C: $5$

   D Option D: $9.4$
   Answer and misconceptions

   A Option A: $7$ ✗

   Adds the horizontal and vertical distancesCalculates 3 + 4; the straight-line distance is the hypotenuse, √(3² + 4²).

   B Option B: $25$ ✗

   Forgets to square root at the endFinds 3² + 4² = 25 and stops.

   C Option C: $5$ ✓ Correct

   D Option D: $9.4$ ✗

   Adds coordinates instead of finding differencesCalculates √((1 + 4)² + (2 + 6)²) = √89; it should use 4 − 1 and 6 − 2.
5. pyth-05 Present from here

   Which three lengths could be the sides of a right-angled triangle?

   A Option A: $7, 24, 25$

   B Option B: $4, 5, 6$

   C Option C: $5, 7, 12$

   D Option D: $2, 3, 13$
   Answer and misconceptions

   A Option A: $7, 24, 25$ ✓ Correct

   B Option B: $4, 5, 6$ ✗

   Thinks any consecutive numbers work like 3, 4, 54² + 5² = 41, not 6² = 36.

   C Option C: $5, 7, 12$ ✗

   Checks if the shorter sides add to the longestTests 5 + 7 = 12 instead of 5² + 7² = 12².

   D Option D: $2, 3, 13$ ✗

   Forgets to square root the longest sideWorks out 2² + 3² = 13 and takes 13 as the side; it should be √13.