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We have this shape with those sides.
We can draw in a circle so it touches those 3 sides.
Find b and d.

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?

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**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.

We have this shape with those sides. We can draw in a circle so it touches those 3 sides. Find b and d.

Explore a geometry problem involving a right triangle with an inscribed circle that touches three sides. Given values for a and c, learn how to find the unknowns b and d using tangent properties and the Pythagorean theorem. Suitable for high school students in Grades 9-10.

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