Question

6 rajah 3 menunjukkan keratan rentas pintu hadapan sebuah bangsal di mana abcd adalah sebuah segi empat tepat dan oapb adalah sebuah sektor berpusat di o dengan jejari r m. diagram 3 shows the cross section of a front door of a barn such that abcd is a rectangle and oapb is a sector with centre o of radius r m. rajah 3 / diagram 3 pq adalah pembahagi dua sama serenjang ab. diberi bahawa panjang ab ialah 8 m dan panjang pq ialah 2 m. pq is a perpendicular bisector ab. given that the length ab is 8 m and pq is 2 m. guna / use : π = 3.142 (a) hitung nilai r, dalam m, calculate the value of r, in m,

Explanation:

Step1: Find the length of \( AQ \)

Since \( PQ \) is the perpendicular bisector of \( AB \), \( AQ=\frac{AB}{2} \). Given \( AB = 8\ m \), then \( AQ=\frac{8}{2}=4\ m \). Let \( OQ=r - 2\) (because \( OP = r\) and \( PQ=2\)).

Step2: Apply the Pythagorean theorem in right - triangle \( OAQ \)

In right - triangle \( OAQ \), by the Pythagorean theorem \(OA^{2}=AQ^{2}+OQ^{2}\). Since \( OA=r \), we have the equation \(r^{2}=4^{2}+(r - 2)^{2}\).
Expand \((r - 2)^{2}\) using the formula \((a - b)^{2}=a^{2}-2ab + b^{2}\), so \(r^{2}=16+r^{2}-4r + 4\).

Step3: Solve the equation for \( r \)

Subtract \(r^{2}\) from both sides of the equation \(r^{2}=16+r^{2}-4r + 4\). We get \(0=20-4r\).
Then, \(4r=20\), and \(r = 5\).

Answer:

\(r = 5\ m\)