ABCD is a right-angled trapezium at B in which $$ \overrightarrow{A D} / / \overrightarrow{B C}, \mathrm{AB}=8 \mathrm{~cm} $$, $$ \mathrm{BC}=15 \mathrm{~cm} $$ and $$ \mathrm{AD}=9 \mathrm{~cm} $$. Forces of magnitudes $$ \mathrm{F}, 44 $$ and $$ 68 \mathrm{gm} . \mathrm{wt} $$ act in $$ \overrightarrow{D A}, \overrightarrow{D C} $$ and $$ \overrightarrow{A C} $$ respectively. If the line of action of the resultant passes through B , Then the value of $$ \mathrm{F}=\ldots \ldots . $$. gm.wt. $$ \begin{array}{llll} ext { (a) } 114 & ext { (b) } 126 & ext { (c) } 156 & ext { (d) } 184\end{array} $$

Question

ABCD is a right-angled trapezium at B in which $$ \overrightarrow{A D} / / \overrightarrow{B C}, \mathrm{AB}=8 \mathrm{~cm} $$, $$ \mathrm{BC}=15 \mathrm{~cm} $$ and $$ \mathrm{AD}=9 \mathrm{~cm} $$. Forces of magnitudes $$ \mathrm{F}, 44 $$ and $$ 68 \mathrm{gm} . \mathrm{wt} $$ act in $$ \overrightarrow{D A}, \overrightarrow{D C} $$ and $$ \overrightarrow{A C} $$ respectively. If the line of action of the resultant passes through B , Then the value of $$ \mathrm{F}=\ldots \ldots . $$. gm.wt. $$ \begin{array}{llll}	ext { (a) } 114 & 	ext { (b) } 126 & 	ext { (c) } 156 & 	ext { (d) } 184\end{array} $$

UpStudy AI · Accepted Answer

To determine the value of $$ \mathrm{F} $$ in the given right-angled trapezium $$ ABCD $$ with a right angle at $$ B $$, follow these steps:

### Step 1: Understand the Geometry
- **Vertices Coordinates**:
  - Let $$ B $$ be at the origin $$(0,0)$$.
  - $$ A $$ is at $$(0,8)$$ since $$ \mathrm{AB} = 8 $$ cm.
  - $$ C $$ is at $$(15,0)$$ because $$ \mathrm{BC} = 15 $$ cm.
  - $$ D $$ must be at $$(9,8)$$ because $$ \mathrm{AD} = 9 $$ cm and $$ \overrightarrow{AD} \parallel \overrightarrow{BC} $$.

### Step 2: Identify the Forces and Their Lines of Action
- **Forces**:
  - **$$ \mathrm{F} $$** acts along $$ \overrightarrow{DA} $$: Direction is horizontal leftward.
  - **44 gm wt** acts along $$ \overrightarrow{DC} $$: Direction is from $$ D(9,8) $$ to $$ C(15,0) $$.
  - **68 gm wt** acts along $$ \overrightarrow{AC} $$: Direction is from $$ A(0,8) $$ to $$ C(15,0) $$.

### Step 3: Calculate Moments About Point $$ B $$
- **Moment of $$ \mathrm{F} $$ (Clockwise)**:
  $$
  \text{Moment}_F = F \times 8
  $$
  (since the perpendicular distance from $$ B $$ to $$ \overrightarrow{DA} $$ is 8 cm)

- **Moment of 44 gm wt (Counterclockwise)**:
  $$
  \text{Moment}_{44} = 44 \times 12
  $$
  (perpendicular distance calculated using the line equation and distance formula)

- **Moment of 68 gm wt (Counterclockwise)**:
  $$
  \text{Moment}_{68} = 68 \times 7.06
  $$
  (again, using the distance formula)

### Step 4: Equilibrium Condition
For the resultant force to pass through $$ B $$, the sum of the moments about $$ B $$ must be zero:
$$
8F - 528 - 480 = 0
$$
$$
8F = 1008 \implies F = 126 \text{ gm wt}
$$

### Final Answer
**Option (b) 126**
$$ F = 126 $$ gm.wt

ABCD is a right-angled trapezium at B in which ,
and . Forces of magnitudes and act in
and respectively. If the line of action of the resultant passes
through B , Then the value of . gm.wt.

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ABCD is a right-angled trapezium at B in which , and . Forces of magnitudes and act in and respectively. If the line of action of the resultant passes through B , Then the value of . gm.wt.