Track quiz

Twelve ideas, twelve checks. The table comes first, so nothing has to be remembered cold.

Twelve facts about right angles, roots and triples
Idea	How it goes
The long side	Always across from the square corner, however the triangle is drawn.
Its name	The hypotenuse. The other two run into the square corner.
A square on a side	A side of length s carries a square of s × s tiles.
The relationship	On a right-angled triangle, a² + b² = c². 9 + 16 = 25.
Only at 90°	Under 90°, the two at the corner are bigger. Over 90°, the one across is bigger.
Long side wanted	Square both short sides, add, then hunt for a match. It may sit between two whole numbers.
Short side wanted	Subtract the known short square from the long square, then hunt.
Testing three lengths	Do they close? Square the two smaller and add. One apart is still no.
Equal sides	Take a right triangle with two equal sides a. Its long side has c² = 2a².
A root as a decimal	Square your guess. Too low, guess higher. Too high, guess lower.
Scaled triples	Multiply every side of a triple by the same whole number. It is still a triple. A triple with no common factor above 1 is primitive.
A hidden depth	The bent stem keeps its length. Write the sides of the right triangle with d, then solve.

A right-angled triangle has corners P, Q and R. The square corner is at Q. Which side is the longest?

PQ, because it is written first
PR
It depends how the triangle is drawn

A square is built outwards on a side of length 12. How many unit tiles does it hold?

24, since 12 doubled is 24
144
48, one row of 12 on each of four sides

A right-angled triangle has short sides of 15 and 20. What does the square on its long side hold?

35, because 15 + 20 is 35
625
300, because 15 × 20 is 300

Two sides meet at one corner. Their squares hold 144 and 97 tiles. The square across from that corner holds 337. What does that say about the corner?

It is square — 144 and 97 are close enough to 337
It's over 90° — 144 + 97 is 241, and 337 is bigger
The rule always holds, so 337 must be a measuring error

A right-angled triangle has short sides of 2 and 3. What can you say exactly about the long side?

It is 5, because 2 + 3 is 5
Its square is 13. Since 9 < 13 < 16, the side sits between 3 and 4
Its square is 13, so the side is 13 divided by 2

A 25 m rope runs from a pole's top to a peg. The peg sits 7 m from the foot. How tall is the pole?

18 m, because 25 − 7 is 18
24 m
26 m, from adding instead

Three sticks measure 8, 9 and 12. Do they make a right-angled triangle?

Yes — 145 and 144 are one apart, close enough
No. 64 + 81 is 145, and 12 × 12 is 144. One apart is not equal
There is no triangle, because 8 + 9 is more than 12

A gate panel is 20 cm across and 21 cm high. A diagonal bar runs corner to corner. How long is the bar, and why is that an adding job?

41 cm, because 20 + 21 is 41
29 cm. The bar faces the square corner. Both sides are short
Between 6 and 7 cm, from subtracting 400 from 441

A right triangle has two equal sides of 20. What is c²?

1.7 × 1.7 = 2.89 and 1.8 × 1.8 = 3.24. What is true of the side whose square is 3?

It is between 1.7 and 1.8
It is exactly 1.75
It is exactly 1.8

(15, 20, 25) is a triple. Is (45, 60, 75) a triple?

No, the sides must be the smallest numbers
No, because 45 + 60 is more than 75
Yes, every side is 3 times bigger

A reed stands 2 cm above a pond. Bent, its tip touches the water 8 cm away. How deep is the pond?

15 cm
17 cm
23 cm

Chapter 2 of Part II is The Baudhayana-Pythagoras Theorem. You have counted tiles and swung a corner off 90°. You have tested a gate with a tape. See what the chapter asks of you now.

Next: Proportional Reasoning-2. It has pie charts. One amount rises as another falls.

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