Short Answer 
To find missing lengths in a right-angled triangle, apply the Pythagorean theorem and set up equations based on the sides of the triangles. After forming and solving the equations, you can calculate the lengths of segments, resulting in approximate values such as AH (6.75 units), BH (6.28 units), and HC (5.25 units).
Step 1: Apply the Pythagorean Theorem
To find missing lengths in a right-angled triangle, start by applying the Pythagorean theorem. For a triangle ABC with AC as the hypotenuse, the theorem states:
- ∣AC∣² = ∣AB∣² + ∣BC∣²
In your case, you will use this theorem to set up equations that relate the sides of triangles AHB and BHC. Make sure to define lengths for the segments appropriately, such as letting AH be x units and BH be y units.
Step 2: Form and Solve the Equations
Next, set up the equations based on the triangles. For triangle BHC, the equation becomes:
- ∣BC∣² = ∣HC∣² + ∣BH∣²
By substituting known lengths, you’ll produce a system of equations. For triangle AHB, you also form:
- ∣AH∣² + ∣BH∣² = 9²
Solve the system using methods such as substitution or elimination to find the unknowns x and y.
Step 3: Calculate the Lengths
After solving the equations, you can determine the lengths of the segments. For instance, once x and y are calculated:
- Length of AH (x) ≈ 6.75 units
- Length of BH (y) ≈ 6.28 units
- Length of HC = 12 Рx ≈ 5.25 units
Thus, you conclude that the length of the line segment HC is approximately 5.25 units. This systematic approach ensures you accurately find the missing lengths in right-angled triangles.