Question

4 rajah menunjukkan seke ping papan berbentuk segi empat tepat. seorang pekerja ingin memotong papan itu kepada dua keping papan berbentuk segi tiga. perimeter segi tiga ialah 24 cm dan ukuran sisi terpanjang segi tiga itu ialah (x + y) cm. hitung luas, dalam cm², (jwp : 48) 6 markah

Explanation:

Step1: Recall perimeter formula for right - triangle

For the right - triangle formed from the rectangle, the sides are \(3x\), \(y\) and \(x + y\). The perimeter \(P=3x + y+(x + y)=24\).
Simplify the perimeter equation:
\[3x + y+x + y=24\]
\[4x + 2y=24\]
\[2x + y = 12\]
\[y=12 - 2x\]

Step2: Apply Pythagorean theorem

By the Pythagorean theorem, \((3x)^{2}+y^{2}=(x + y)^{2}\).
Expand the equation:
\[9x^{2}+y^{2}=x^{2}+2xy + y^{2}\]
Subtract \(y^{2}\) from both sides:
\[9x^{2}=x^{2}+2xy\]
\[9x^{2}-x^{2}-2xy = 0\]
\[8x^{2}-2xy = 0\]
Factor out \(2x\):
\[2x(4x - y)=0\]
Since \(x
eq0\) (if \(x = 0\), the figure degenerates), then \(4x-y = 0\), so \(y = 4x\).

Step3: Solve for \(x\)

Substitute \(y = 4x\) into \(y=12 - 2x\):
\[4x=12 - 2x\]
Add \(2x\) to both sides:
\[4x+2x=12\]
\[6x=12\]
\[x = 2\]

Step4: Solve for \(y\)

Substitute \(x = 2\) into \(y = 4x\), we get \(y=8\).

Step5: Calculate the area of the rectangle

The area of the rectangle \(A=3x\times y\).
Substitute \(x = 2\) and \(y = 8\) into the area formula:
\[A=3\times2\times8\]
\[A = 48\]

Answer:

48