The diagram shows a circle, centre $O$, diameter $AB.$
$A,B$ and $C$ lie on the circumference of the circle.
(a) Write down the mathematical name of the line $AC.$
(b) Find the value of $x.$
Give a geometrical reason for your answer.
▶️ Answer/Explanation
(a) Ans: Chord
The line \( AC \) is called a chord because it is a straight line segment connecting two points (\( A \) and \( C \)) on the circumference of the circle.
(b) Ans: \( 58 \)
Since \( AB \) is the diameter and \( C \) lies on the circumference, the angle \( ACB \) is \( 90^\circ \) (angle in a semicircle). In triangle \( ABC \), the sum of angles is \( 180^\circ \). Given \( \angle CAB = 32^\circ \), we solve for \( x \) as follows:
\[ x = 180^\circ – 90^\circ – 32^\circ = 58^\circ \]
Geometrical reason: The angle subtended by the diameter in a semicircle is a right angle (\( 90^\circ \)).
Ravi cycles from home to the bank.
It takes him 15 minutes, cycling at a constant speed of 14km/h.
(a) Work out how far Ravi cycles from home to the bank.
(b) Ravi stays at the bank for 18 minutes.
He then cycles home at a constant speed for 12 minutes.
Draw the travel graph to show Ravi’s journey since he left home.
▶️ Answer/Explanation
(a) Ans: 3.5 km
Given speed = 14 km/h and time = 15 minutes. Convert time to hours: \( \frac{15}{60} = 0.25 \) hours.
Distance = Speed × Time = \( 14 \times 0.25 = 3.5 \) km.
(b)
Graph description: Starts at (0, 0), rises to (15, 3.5) for cycling to bank, flat from (15, 3.5) to (33, 3.5) for stay, then falls to (45, 0) for return trip.
Graph image:
