In `DeltaABC`, draw `BE_|_AC `and `CF _|_ AB` and the perpendicular BE and CF Intersect at the point O. If `/_BAC = 70^@`, then the value of `/_BOC `is

Options

A. `125^@`
B. `55^@`
C. `150^@`
D. `110^@`

Answer

To solve the problem step by step, we will analyze the triangle \( \Delta ABC \) and the perpendiculars drawn from points \( B \) and \( C \).

### Step 1: Understand the Triangle and Perpendiculars
We have triangle \( \Delta ABC \) with \( \angle BAC = 70^\circ \). We draw \( BE \perp AC \) and \( CF \perp AB \), where \( E \) is the foot of the perpendicular from \( B \) to \( AC \) and \( F \) is the foot of the perpendicular from \( C \) to \( AB \). These two perpendiculars intersect at point \( O \).

**Hint:** Visualize the triangle and the perpendiculars to understand the relationships between the angles.

### Step 2: Identify Known Angles
From the problem, we know:
- \( \angle BAC = 70^\circ \)
- \( \angle BEA = 90^\circ \) (since \( BE \perp AC \))
- \( \angle CFA = 90^\circ \) (since \( CF \perp AB \))

**Hint:** Remember that the angles formed by perpendicular lines are always \( 90^\circ \).

### Step 3: Analyze Quadrilateral \( AODE \)
In quadrilateral \( AODE \), we can use the property that the sum of the angles in a quadrilateral is \( 360^\circ \):
\[ \angle A + \angle O + \angle E + \angle D = 360^\circ \]
Substituting the known values:
\[ 70^\circ + \angle O + 90^\circ + 90^\circ = 360^\circ \]

**Hint:** Set up the equation carefully by substituting the known angles.

### Step 4: Simplify the Equation
Now simplify the equation:
\[ 70^\circ + 90^\circ + 90^\circ = 250^\circ \]
Thus, we have:
\[ 250^\circ + \angle O = 360^\circ \]

**Hint:** Combine the angles on the left side to find the unknown angle.

### Step 5: Solve for \( \angle O \)
Now, isolate \( \angle O \):
\[ \angle O = 360^\circ - 250^\circ = 110^\circ \]

**Hint:** Subtract to find the value of the angle.

### Step 6: Relate \( \angle O \) to \( \angle BOC \)
Since \( \angle BOC \) is vertically opposite to \( \angle O \), we have:
\[ \angle BOC = \angle O = 110^\circ \]

**Hint:** Remember that vertically opposite angles are equal.

### Conclusion
Therefore, the value of \( \angle BOC \) is \( 110^\circ \).

**Final Answer:** \( \angle BOC = 110^\circ \)