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Exercise 1:

1. Correct the fractions:

Rewrite the given fractions with simplified or correct numerical expressions.

• $\frac{3}{\sqrt{3}} = \sqrt{3}$ (Rationalized)
• $\frac{6\sqrt{5}}{2\sqrt{3}} = \frac{3\sqrt{5}}{\sqrt{3}} = \sqrt{15}$ (Simplified)
• $\frac{2\sqrt{2}}{\sqrt{2} - 1}$: Multiply numerator and denominator by $(\sqrt{2} + 1)$: $= \frac{2\sqrt{2}(\sqrt{2} + 1)}{(\sqrt{2} - 1)(\sqrt{2} + 1)} = \frac{4 + 2\sqrt{2}}{1} = 4 + 2\sqrt{2}.$
• $\frac{\sqrt{2}}{2 + \sqrt{3}}$: Multiply numerator and denominator by $(2 - \sqrt{3})$: $= \frac{\sqrt{2}(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})} = \frac{2\sqrt{2} - \sqrt{6}}{1} = 2\sqrt{2} - \sqrt{6}.$

2. Solve the equations:

• $x^2 + 5 = 2$: $x^2 = -3 \implies \text{No real solution (complex roots)}.$
• $(x - 5)^2 = 9$: $x - 5 = \pm 3 \implies x = 8 \; \text{or} \; x = 2.$
• $x^2 + 3 = 9$: $x^2 = 6 \implies x = \pm \sqrt{6}.$

Exercise 2:

Geometry Problem:

Given $ABCD$ with specific dimensions:

• Use coordinate geometry or Pythagorean Theorem to solve the relationships between segments.

Steps:

1. Compute $\overrightarrow{DE}$, $\overrightarrow{AF}$, etc.
2. Prove $(AD) \parallel (NF)$ by showing proportional relationships or using slope calculations.

Exercise 3:

1. Compute $DK$ and $AK$:

Use the Pythagorean Theorem or trigonometric functions based on the given diagram.

2. Compute $DB$:

Use the diagonal relationship or trigonometric functions to solve for $DB$.

3. Prove $BDC$ is a right triangle:

Use the Pythagorean Theorem: $BC^2 + DC^2 = BD^2.$

4. Compute ratios of triangles $ABD$ and $BCD$:

Calculate areas or similar proportions of triangles using their side lengths.

Would you like me to solve a specific part in detail or explain any step further?

Here are five related questions you might find helpful:

1. How do you rationalize a fraction with square roots?
2. What is the geometric proof of parallel lines in a quadrilateral?
3. How to apply trigonometry to calculate triangle heights or bases?
4. Can you verify if two triangles are similar or congruent?
5. How to derive the area ratio of two triangles sharing a common side?

Tip: Always verify the units and dimensions when working with geometry problems to avoid mistakes.