Hint
Use the stated units and precision. Use the perpendicular chord bisector, apply Pythagoras, then double.
Worked solution
(a)
Use the perpendicular chord bisector, apply Pythagoras, then double.
The half-chord is \(\displaystyle \sqrt{r^{2}-d^{2}}=8 \mathrm{cm}\). Double it to obtain 16 cm.
Answer: \(16\)
(b)
Use the tangent-radius right angle and OP as hypotenuse.
\(\displaystyle OT\perp PT\), so \(\displaystyle PT=\sqrt{676-100}=24 \mathrm{cm}\).
Answer: \(24\)
(c)
Use equality of the two tangents from the same external point.
Tangents from P are equal, so \(\displaystyle PU=PT=24 \mathrm{cm}\).
Answer: \(24\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternative methods and consequential reasoning. An unexplained result does not earn a method mark.
(a) · 2 marks
- 1 mark: Use the perpendicular chord bisector, apply Pythagoras, then double.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Keep intermediate values exact until the final step.
(b) · 2 marks
- 1 mark: Use the tangent-radius right angle and OP as hypotenuse.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Check the answer against the original conditions.
(c) · 2 marks
- 1 mark: Use equality of the two tangents from the same external point.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Keep intermediate values exact until the final step.
Total: 6 marks.