Question

In triabgle ABC, C is the midpoint of BD, AB=10cm ,AD=12 cm AC=9cm find BD ?

• A. 2√35
• B. 2√50
• C. 2√65
• D. 2√41

Answer 1

Answer:

D. 2√41

Apollonius' Theorem

• Apollonius' Theorem states that for any triangle, the sum of the squares of two sides is equal to twice the sum of the square of the median to the third side and the square of half of the third side.

• In Triangle ABD, AC is the median to side BD because C is the midpoint of BD.

• The formula for Apollonius' Theorem in triangle ABD with median AC is:AB² + AD² = 2(AC² + CD²)or equivalently, AB² + AD² = 2(AC² + BC²) (since C is the midpoint, BC = CD).

Step-by-Step Calculation

1. Identify the given values:

   • AB = 10 cm

   • AD = 12 cm

   • AC = 9 cm

   • C is the midpoint of BD.

2. Apply Apollonius' Theorem:

   • Substitute the given values into the formula: 10² + 12² = 2(9² + CD²)

3. Calculate the squares:

   • 100 + 144 = 2(81 + CD²)

4. Simplify the equation:

   • 244 = 162 + 2CD²

5. Isolate 2CD²:

   • 2CD² = 244 - 162

   • 2CD² = 82

6. Solve for CD²:

   • CD² = 82 / 2

   • CD² = 41

7. Find CD:

   • CD = √41 cm

8. Determine BD:

   • Since C is the midpoint of BD, BD = 2 * CD.

   • BD = 2 * √41 cm.