Master the intersection of Algebra and Geometry: Lines, Slopes, and Shapes on the Cartesian Plane.
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<p><strong>Quadrants:</strong></p>
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<li><strong>Q1 (+, +)</strong>: Top Right</li>
<li><strong>Q2 (-, +)</strong>: Top Left</li>
<li><strong>Q3 (-, -)</strong>: Bottom Left</li>
<li><strong>Q4 (+, -)</strong>: Bottom Right</li>
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<h4 class='font-bold text-primary-700 mb-1'>Distance Formula</h4>
<p>Distance between $A(x_1, y_1)$ and $B(x_2, y_2)$:</p>
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$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
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<h5 class='font-bold text-primary-700 mb-2'>Internal Division (Ratio m:n)</h5>
<p>$P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)$</p>
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<h5 class='font-bold text-primary-700 mb-2'>Midpoint (Ratio 1:1)</h5>
<p>$M(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$</p>
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<h5 class='font-bold text-primary-700 mb-1'>Centroid of Triangle</h5>
<p>Intersection of Medians:</p>
<p class='text-center font-bold'>$G(x, y) = \left( \frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3} \right)$</p>
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<p>For vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$:</p>
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<p class='font-bold text-lg'>Area = $\frac{1}{2} | x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) |$</p>
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<p class='text-sm text-gray-500'><strong>Note:</strong> If Area = 0, points are <strong>Collinear</strong> (Lie on the same line).</p>
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<strong>Slope ($m$):</strong> $m = \tan \theta = \frac{y_2 - y_1}{x_2 - x_1}$
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<strong>Equation (Slope-Point):</strong> $y - y_1 = m(x - x_1)$
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<strong>Standard Form:</strong> $ax + by + c = 0$ $\to$ Slope $= -a/b$
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<div class='text-center p-2 bg-primary-100 rounded text-primary-800 font-bold'>Parallel Lines<br/>$m_1 = m_2$</div>
<div class='text-center p-2 bg-red-100 rounded text-red-800 font-bold'>Perpendicular<br/>$m_1 \times m_2 = -1$</div>
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