Thank you for the image. From the given diagram and the details provided, triangle $\triangle ABD$ appears to be a right triangle with $\angle BAD = 90^\circ$, and segment $ABC$ forms a straight line. Here are the known values:

• $AC = 3.75 \, \text{cm}$,
• $BC = 4.25 \, \text{cm}$,
• $AB = 2.5 \, \text{cm}$,
• $\angle BAC = 90^\circ$.

To solve this, let us confirm the lengths and geometry relationships step by step.

Step 1: Use Pythagoras' Theorem for $\triangle ABD$

Since $\triangle ABD$ is a right triangle: $AB^2 + BD^2 = AD^2$ Substituting $AB = 2.5 \, \text{cm}$ and $AD = AC = 3.75 \, \text{cm}$: $2.5^2 + BD^2 = 3.75^2$ $6.25 + BD^2 = 14.0625$ $BD^2 = 14.0625 - 6.25 = 7.8125$ $BD = \sqrt{7.8125} \approx 2.795 \, \text{cm}.$

Step 2: Calculate $DC$

Since $BC = 4.25 \, \text{cm}$ and $BD \approx 2.795 \, \text{cm}$, we use: $BC = BD + DC$ $DC = BC - BD = 4.25 - 2.795 \approx 1.455 \, \text{cm}.$

Summary of Results:

• $BD \approx 2.795 \, \text{cm}$,
• $DC \approx 1.455 \, \text{cm}$.

Would you like further details, or do you want me to verify or explore additional aspects of this problem?

Related Questions:

1. How is the Pythagorean theorem applied in different geometries?
2. Can we verify this calculation using a scaled diagram or trigonometric ratios?
3. How to find the area of $\triangle ABD$ and $\triangle BDC$?
4. What are the proportions of the segments $BD$ and $DC$?
5. How would the calculation change if the triangle's dimensions were different?

Tip: In geometry, always double-check relationships between angles and sides, especially in diagrams with multiple triangles.