To solve the problem, we need to analyze the two statements provided and determine their validity step by step.
**Step 1: Analyze Statement I**
- We have a quadrilateral ABCD with the following properties:
- \( \angle C = 90^\circ \)
- \( AB = 9 \, \text{m} \)
- \( BC = 12 \, \text{m} \)
- \( CD = 5 \, \text{m} \)
- \( AD = 8 \, \text{m} \)
**Step 2: Calculate the area of triangle BCD**
- Since \( \angle C = 90^\circ \), we can use the formula for the area of a right triangle:
\[
\text{Area}_{BCD} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times BC \times CD = \frac{1}{2} \times 12 \times 5
\]
\[
= \frac{1}{2} \times 60 = 30 \, \text{m}^2
\]
**Step 3: Calculate the length of diagonal BD using Pythagorean theorem**
- In triangle BCD, we can find \( BD \):
\[
BD^2 = BC^2 + CD^2 = 12^2 + 5^2 = 144 + 25 = 169
\]
\[
BD = \sqrt{169} = 13 \, \text{m}
\]
**Step 4: Calculate the area of triangle ABD using Heron's formula**
- First, we find the semi-perimeter \( s \):
\[
s = \frac{AB + AD + BD}{2} = \frac{9 + 8 + 13}{2} = \frac{30}{2} = 15 \, \text{m}
\]
- Now, we apply Heron's formula:
\[
\text{Area}_{ABD} = \sqrt{s(s - AB)(s - AD)(s - BD)} = \sqrt{15(15 - 9)(15 - 8)(15 - 13)}
\]
\[
= \sqrt{15 \times 6 \times 7 \times 2}
\]
\[
= \sqrt{15 \times 84} = \sqrt{1260}
\]
- Approximating \( \sqrt{1260} \) gives us about \( 35.5 \, \text{m}^2 \).
**Step 5: Calculate the total area of quadrilateral ABCD**
- The total area of quadrilateral ABCD is:
\[
\text{Area}_{ABCD} = \text{Area}_{ABD} + \text{Area}_{BCD} = 35.5 + 30 = 65.5 \, \text{m}^2
\]
**Step 6: Validate Statement I**
- Statement I claims the area is \( 70 \, \text{m}^2 \), which is incorrect since we calculated it to be \( 65.5 \, \text{m}^2 \).
**Step 7: Analyze Statement II**
- The sides of the triangle are \( 13 \, \text{cm}, 14 \, \text{cm}, 15 \, \text{cm} \).
- Calculate the semi-perimeter:
\[
s = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21 \, \text{cm}
\]
- Statement II is true.
**Final Conclusion**
- Statement I is false, and Statement II is true.
**Correct Option: Statement 1 is false, Statement 2 is true.**
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