https://file.aiquickdraw.com/mathbot/file/1727403716934msj1mbzy.jpg

The problem involves the triangle \( \triangle ACD \) with given conditions: 

- \( BD = 5 \),
- \( \angle DAB = 45^\circ \),
- \( AB:BC = 1:4 \).

You are asked to calculate the area of \( \triangle DAC \).

### Approach

1. **Understanding the triangle**: 
   From the diagram, the angle \( \angle DAB \) is 45°, and we are given \( BD = 5 \). Based on this information, we can calculate the area of \( \triangle DAC \).

2. **Using the Area Formula**:
   The area of a triangle can be calculated using the formula:
   \[
   \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
   \]
   From the diagram, it appears that the base of \( \triangle DAC \) is \( BD = 5 \), and the height (from point \( A \)) is 5 (since \( AB:BC = 1:4 \), this simplifies the situation).

3. **Area Calculation**:
   Substituting the values into the area formula:
   \[
   \text{Area} = \frac{1}{2} \times 5 \times 25 = \frac{1}{2} \times 125 = 62.5
   \]
   Therefore, the area of \( \triangle DAC \) is \( 62.5 \) square units.

Thus, the correct answer is **(b) 62.5**.

---

Do you want further details on the calculation, or do you have any questions?

Here are some related questions for further understanding:

1. How would the area change if \( BD \) was doubled?
2. What would be the area if \( \angle DAB \) was 60° instead of 45°?
3. How can you use trigonometric functions to verify the result?
4. What is the significance of the \( AB:BC = 1:4 \) ratio in this problem?
5. How does knowing the altitude of a triangle affect area calculations?

**Tip**: Always ensure the base and height are perpendicular when using the area formula for triangles!

In triangle ACD, BD = 5, ∠DAB = 45°, and AB:BC = 1:4. The area of triangle DAC is:

Learn how to calculate the area of triangle DAC where BD = 5, ∠DAB = 45°, and the ratio AB:BC = 1:4. Follow the detailed steps involving the area formula and geometric ratios, suitable for Grade 8 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation