Question : The graph of $3x+4y-24=0$ forms a $ riangle OAB$ with the coordinate axes, where $O$ is the origin. Also, the graph of $x+y+4=0$ forms a $ riangle OCD$ with the coordinate axes. Then the area of $ riangle OCD$ is equal to:
Option 1: $\frac{1}{2} ext{ of the area of}\ riangle OAB$
Option 2: $\frac{1}{3} ext{ of the area of}\ riangle OAB$
Option 3: $\frac{2}{3} ext{ of the area of}\ riangle OAB$
Option 4: $ ext{ the area of}\ riangle OAB$

Question

Question : The graph of $3x+4y-24=0$ forms a $ riangle OAB$ with the coordinate axes, where $O$ is the origin. Also, the graph of $x+y+4=0$ forms a $ riangle OCD$ with the coordinate axes. Then the area of $ riangle OCD$ is equal to:

Option 1: $\frac{1}{2} ext{ of the area of}\ riangle OAB$

Option 2: $\frac{1}{3} ext{ of the area of}\ riangle OAB$

Option 3: $\frac{2}{3} ext{ of the area of}\ riangle OAB$

Option 4: $ ext{ the area of}\ riangle OAB$

Team Careers360 · Accepted Answer

Correct Answer: $\frac{1}{3} ext{ of the area of}\ riangle OAB$

Solution :
Given: The graph of $3x+4y-24=0$ forms a $ riangle OAB$ with the coordinate axes, where $O$ is the origin. Also, the graph of $x+y+4=0$ forms a $ riangle OCD$ with the coordinate axes.
By substituting the value of $x=0$ in the equation $3x + 4y= 24$, we get,
$3 imes0+ 4y= 24$
⇒ $4y=24$
⇒ $y = 6$
Therefore, B's coordinates are $(0, 6)$.
By substituting the value of $y=0$ in the equation $3x + 4y = 24$, we get,
$3x + 4 imes 0 = 24$
⇒ $3x=24$
⇒ $x = 8$
Therefore, A's coordinates are $(8,0)$.
By substituting the value of $y=0$ in the equation $x + y = –4$, we get,
$x + 0 = –4$
⇒ $x=–4$
Therefore, C's coordinates are $(–4,0 )$.
By substituting the value of $x=0$ in the equation $x + y = –4$, we get,
$0 + y = –4$
⇒ $y=–4$
Therefore, D's coordinates are $(0,–4)$.
Also, the area of $ riangle OAB=\frac{1}{2} imes OA imes OB$.
⇒ $\frac{1}{2} imes 8 imes 6=24$ sq. units
Similarly, the area of $ riangle OCD=\frac{1}{2} imes OC imes OD$.
⇒ $\frac{1}{2} imes 4 imes 4=8$ sq. units
So, $ ext{ area of}\ riangle OCD=\frac{1}{3} $ of the area of $ riangle OAB$
Hence, the correct answer is $\frac{1}{3} ext{ of the area of}\ riangle OAB$.

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