Master the intersection of Algebra and Geometry: Lines, Slopes, and Shapes on the Cartesian Plane.
<div class='space-y-4'>
<p><strong>Quadrants:</strong></p>
<ul class='list-disc pl-5 space-y-1 text-gray-600'>
<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>
</ul>
<div class='bg-primary-50 p-4 rounded-lg border-l-4 border-primary-500'>
<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>
<div class='text-center text-lg font-bold text-gray-800 my-2'>
$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
</div>
</div>
</div>
<div class='space-y-6'>
<div class='grid grid-cols-1 md:grid-cols-2 gap-4'>
<div class='bg-white p-4 rounded-xl border border-gray-200 shadow-sm'>
<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>
</div>
<div class='bg-white p-4 rounded-xl border border-gray-200 shadow-sm'>
<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>
</div>
</div>
<div class='bg-primary-50 p-4 rounded-lg border-l-4 border-primary-500'>
<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>
</div>
</div>
<div class='space-y-4'>
<p>For vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$:</p>
<div class='bg-red-50 p-4 rounded-lg text-center border border-red-200'>
<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>
</div>
<p class='text-sm text-gray-500'><strong>Note:</strong> If Area = 0, points are <strong>Collinear</strong> (Lie on the same line).</p>
</div>
<div class='space-y-4'>
<ul class='list-none space-y-3'>
<li class='bg-gray-50 p-3 rounded border-l-4 border-gray-500'>
<strong>Slope ($m$):</strong> $m = \tan \theta = \frac{y_2 - y_1}{x_2 - x_1}$
</li>
<li class='bg-gray-50 p-3 rounded border-l-4 border-gray-500'>
<strong>Equation (Slope-Point):</strong> $y - y_1 = m(x - x_1)$
</li>
<li class='bg-gray-50 p-3 rounded border-l-4 border-gray-500'>
<strong>Standard Form:</strong> $ax + by + c = 0$ $\to$ Slope $= -a/b$
</li>
</ul>
<div class='grid grid-cols-2 gap-4 mt-2'>
<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>
</div>
</div>