In a `DeltaABC`, if `angleA` = `115^@`, `angleC=20^@` and D is a point on BC such that AD perpendicular to BC and BD = 7 cm,then AD is of length

Options

A. 15cm
B. 5 cm
C. 7 cm
D. 10 cm

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the angles in triangle ABC
Given:
- Angle A = 115°
- Angle C = 20°

We can find Angle B using the property that the sum of angles in a triangle is 180°.

**Calculation:**
Angle B = 180° - Angle A - Angle C
Angle B = 180° - 115° - 20°
Angle B = 45°

### Step 2: Analyze triangle ADC
Since D is a point on BC such that AD is perpendicular to BC, we can analyze triangle ADC.

In triangle ADC:
- Angle C = 20°
- Angle D = 90° (since AD is perpendicular to BC)

Using the property of angles in a triangle:
Angle DAC + Angle D + Angle C = 180°

**Calculation:**
Let Angle DAC = x.
x + 90° + 20° = 180°
x + 110° = 180°
x = 180° - 110°
x = 70°

So, Angle DAC = 70°.

### Step 3: Find Angle BAD
Now we can find Angle BAD:
Angle BAD = Angle BAC - Angle DAC
Angle BAD = 115° - 70°
Angle BAD = 45°

### Step 4: Use trigonometric ratios in triangle ADB
In triangle ADB, we can use the tangent function since we have an angle and the opposite side (AD) and adjacent side (BD).

Using the tangent function:
tan(Angle BAD) = Opposite/Adjacent
tan(45°) = AD/BD

Since tan(45°) = 1:
1 = AD/7 cm

**Calculation:**
AD = 7 cm * 1
AD = 7 cm

### Final Answer:
The length of AD is 7 cm.

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