ABC is a triangle in which ∠A=90∘ , AN⊥BC, BC=12 cm and AC=5 cm. Find the ratio of the areas of △ANC and △ABC.

Options

A. 125:44
B. 25:144
C. 144:25
D. 12:5

Answer

To solve the problem, we need to find the ratio of the areas of triangles ANC and ABC. Let's break this down step by step.

### Step 1: Identify the triangles and their dimensions
We have triangle ABC where:
- ∠A = 90°
- BC = 12 cm (the base of triangle ABC)
- AC = 5 cm (one of the legs of triangle ABC)

### Step 2: Calculate the area of triangle ABC
The area of a right triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
In triangle ABC, we can take AC as the height and BC as the base.
\[ \text{Area}_{ABC} = \frac{1}{2} \times BC \times AC = \frac{1}{2} \times 12 \times 5 \]
Calculating this gives:
\[ \text{Area}_{ABC} = \frac{1}{2} \times 60 = 30 \text{ cm}^2 \]

### Step 3: Find the length of AN
Since AN is perpendicular to BC, we can use the Pythagorean theorem to find the length of AN. We need to find the length of AB first. Using the Pythagorean theorem in triangle ABC:
\[ AB^2 + AC^2 = BC^2 \]
Let AB = x. Then:
\[ x^2 + 5^2 = 12^2 \]
\[ x^2 + 25 = 144 \]
\[ x^2 = 144 - 25 = 119 \]
\[ x = \sqrt{119} \text{ cm} \]

### Step 4: Calculate the area of triangle ANC
Triangle ANC is also a right triangle with AN as the height and AC as the base. The area of triangle ANC can be calculated as:
\[ \text{Area}_{ANC} = \frac{1}{2} \times AC \times AN \]
To find AN, we can use the property of similar triangles. Since AN is the height from A to BC, we can find AN using the ratio of the areas:
\[ \text{Area}_{ANC} = \frac{AC}{BC} \times \text{Area}_{ABC} \]
Substituting the values we have:
\[ \text{Area}_{ANC} = \frac{5}{12} \times 30 = \frac{150}{12} = 12.5 \text{ cm}^2 \]

### Step 5: Find the ratio of the areas of triangles ANC and ABC
Now we can find the ratio of the areas:
\[ \text{Ratio} = \frac{\text{Area}_{ANC}}{\text{Area}_{ABC}} = \frac{12.5}{30} = \frac{25}{60} = \frac{5}{12} \]

### Final Answer
The ratio of the areas of triangle ANC to triangle ABC is:
\[ \frac{5}{12} \]