In `DeltaABC, /_C = 54^@`, the perpendicular bisector of AB at D meets BC at E. If `/_EAC = 42^@`, then what is the value (in degrees) of `/_ABC`?

Options

A. 25
B. 42
C. 50
D. 60

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We have triangle ABC with angle C = 54°. The perpendicular bisector of AB at point D meets BC at point E. We also know that angle EAC = 42°.

### Step 2: Use the properties of the perpendicular bisector
Since D is the midpoint of AB and DE is the perpendicular bisector, angle ADB = angle AEB. Therefore, angle AEB can be expressed as the sum of angles A and C:
\[ \text{Angle AEB} = \text{Angle A} + \text{Angle C} \]

### Step 3: Calculate angle AEB
Given that angle EAC = 42° and angle C = 54°, we can write:
\[ \text{Angle AEB} = \text{Angle EAC} + \text{Angle C} = 42° + 54° = 96° \]

### Step 4: Find angle DEB
Since D is the midpoint and DE is the perpendicular bisector, angle DEB is half of angle AEB:
\[ \text{Angle DEB} = \frac{\text{Angle AEB}}{2} = \frac{96°}{2} = 48° \]

### Step 5: Analyze triangle DEB
In triangle DEB, we know:
- Angle D = 90° (because DE is perpendicular to AB)
- Angle E = 48°

Using the triangle sum property:
\[ \text{Angle D} + \text{Angle E} + \text{Angle B} = 180° \]
Substituting the known values:
\[ 90° + 48° + \text{Angle B} = 180° \]

### Step 6: Solve for angle B
Now, we can solve for angle B:
\[ \text{Angle B} = 180° - 90° - 48° = 42° \]

### Conclusion
Thus, the value of angle ABC is:
\[ \text{Angle ABC} = 42° \]