Find the altitude and base of an isosceles triangle whose vertical angle is `65^(@)` and whose equal sides are 415 cm :

Options

A. 350 cm, 646 cm
B. 350 cm, 446 cm
C. 630 cm, 445 cm
D. none of these

Answer

To solve the problem of finding the altitude and base of an isosceles triangle with a vertical angle of \(65^\circ\) and equal sides of \(415 \, \text{cm}\), we can follow these steps:

### Step 1: Determine the Angles of the Triangle
Since the triangle is isosceles, the two base angles are equal. The sum of the angles in a triangle is \(180^\circ\). Therefore, we can find the base angles as follows:

\[ \text{Base angles} = \frac{180^\circ - 65^\circ}{2} = \frac{115^\circ}{2} = 57.5^\circ \]

### Step 2: Set Up the Triangle
Let’s denote the triangle as \(ABC\) where \(A\) is the vertex with the angle of \(65^\circ\), and \(B\) and \(C\) are the other two vertices. The equal sides \(AB\) and \(AC\) are both \(415 \, \text{cm}\). The altitude \(AD\) from vertex \(A\) to the base \(BC\) will bisect the base \(BC\) into two equal parts, each of length \(\frac{x}{2}\), where \(x\) is the length of the base \(BC\).

### Step 3: Use Trigonometric Ratios to Find the Altitude
Using the sine function for angle \(57.5^\circ\):

\[ \sin(57.5^\circ) = \frac{\text{Altitude (AD)}}{\text{Hypotenuse (AB)}} \]

Substituting the known values:

\[ \sin(57.5^\circ) = \frac{h}{415} \]

Calculating \(\sin(57.5^\circ\):

\[ \sin(57.5^\circ) \approx 0.8434 \]

Setting up the equation:

\[ 0.8434 = \frac{h}{415} \]

Now, solve for \(h\):

\[ h = 415 \times 0.8434 \approx 350 \, \text{cm} \]

### Step 4: Use the Cosine Function to Find the Base
Now, we can use the cosine function for angle \(57.5^\circ\):

\[ \cos(57.5^\circ) = \frac{\frac{x}{2}}{415} \]

Calculating \(\cos(57.5^\circ\):

\[ \cos(57.5^\circ) \approx 0.5373 \]

Setting up the equation:

\[ 0.5373 = \frac{\frac{x}{2}}{415} \]

Now, solve for \(x\):

\[ \frac{x}{2} = 415 \times 0.5373 \]

Calculating the right side:

\[ \frac{x}{2} \approx 445.7 \]

Thus:

\[ x \approx 2 \times 445.7 \approx 891.4 \, \text{cm} \]

### Final Results
- **Altitude (h)**: \(350 \, \text{cm}\)
- **Base (x)**: \(891.4 \, \text{cm}\)