To solve this problem, let's analyze the information given.

1. Given Data:

   • $a = 4$
   • $c = 10$
   • We need to find $b$ and $d$, where $a + c = b + d$.

2. Understanding the Shape:

   • The diagram shows a right triangle with sides $a$, $b$, $c$, and $d$.
   • $a$ and $b$ are perpendicular to each other, and $c$ is the hypotenuse of the triangle.
   • Since a circle is inscribed within the triangle, each side is tangent to the circle. This means we can use the property of tangents to find the unknowns.

3. Using Tangent Properties:

   • For a right triangle with an inscribed circle, the perimeter can be split into segments that are tangents from each vertex to the point where the circle touches each side.
   • The equation $a + c = b + d$ is key here; it relates the sum of two sides to the sum of the other two.

4. Applying the Formula:

   • Since $a = 4$ and $c = 10$, we substitute these values: $4 + 10 = b + d$
   • This simplifies to: $14 = b + d$

5. Using the Pythagorean Theorem:

   • We know that $c$ is the hypotenuse, so: $c = \sqrt{a^2 + b^2}$
   • Substitute $c = 10$ and $a = 4$: $10 = \sqrt{4^2 + b^2}$ $10 = \sqrt{16 + b^2}$ $100 = 16 + b^2$ $b^2 = 84$ $b = \sqrt{84} = 2\sqrt{21} \approx 9.17$

6. Finding $d$:

   • Substitute $b \approx 9.17$ into $b + d = 14$: $d = 14 - 9.17 \approx 4.83$

Final Answer:

• $b \approx 9.17$
• $d \approx 4.83$

Would you like further details or have questions?

Follow-up Questions:

1. How does the inscribed circle relate to the triangle's sides?
2. Can the value of $d$ be confirmed using another approach?
3. How do tangent properties apply in triangles with inscribed circles?
4. What happens if $a + c \neq b + d$?
5. How do we calculate the radius of the inscribed circle here?

Tip: When a circle is inscribed in a triangle, the sum of two opposite sides equals the sum of the other two sides. This property can simplify solving.