`triangle ABC` is a right angled at A and AD is the altitude to BC. If AB= 7 cm and AC = 24 cm. Find the ratio of AD is to AM if M is the mid-point of BC.

Options

A. 25:41
B. 32:41
C. 336/625
D. 625/336

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Identify the triangle and given values
We have triangle ABC, where:
- AB = 7 cm (one leg of the triangle)
- AC = 24 cm (the other leg of the triangle)
- Angle A is the right angle.

### Step 2: Calculate the length of BC using the Pythagorean theorem
Since triangle ABC is a right triangle, we can use the Pythagorean theorem:
\[ BC^2 = AB^2 + AC^2 \]
Substituting the values:
\[ BC^2 = 7^2 + 24^2 = 49 + 576 = 625 \]
Taking the square root:
\[ BC = \sqrt{625} = 25 \text{ cm} \]

### Step 3: Find the midpoint M of BC
Since M is the midpoint of BC, we can find the length of AM:
\[ BM = MC = \frac{BC}{2} = \frac{25}{2} = 12.5 \text{ cm} \]

### Step 4: Calculate the area of triangle ABC
The area of triangle ABC can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times AB \times AC = \frac{1}{2} \times 7 \times 24 = \frac{168}{2} = 84 \text{ cm}^2 \]

### Step 5: Express the area using AD (altitude)
The area can also be expressed using the base BC and height AD:
\[ \text{Area} = \frac{1}{2} \times BC \times AD \]
Substituting the known values:
\[ 84 = \frac{1}{2} \times 25 \times AD \]
Solving for AD:
\[ 84 = 12.5 \times AD \]
\[ AD = \frac{84}{12.5} = \frac{84 \times 2}{25} = \frac{168}{25} \text{ cm} \]

### Step 6: Calculate AM
From Step 3, we found:
\[ AM = \frac{25}{2} = 12.5 \text{ cm} \]

### Step 7: Find the ratio of AD to AM
Now we can find the ratio of AD to AM:
\[ \text{Ratio} = \frac{AD}{AM} = \frac{\frac{168}{25}}{12.5} \]
Converting 12.5 to a fraction:
\[ 12.5 = \frac{25}{2} \]
Now substituting:
\[ \text{Ratio} = \frac{\frac{168}{25}}{\frac{25}{2}} = \frac{168}{25} \times \frac{2}{25} = \frac{336}{625} \]

### Final Result
The ratio of AD to AM is:
\[ \frac{336}{625} \]