{"id":38,"date":"2026-07-09T19:21:43","date_gmt":"2026-07-09T19:21:43","guid":{"rendered":"https:\/\/sqoolpapers.co.za\/notes\/test-analytical-geometry\/"},"modified":"2026-08-22T15:42:35","modified_gmt":"2026-08-22T15:42:35","slug":"analytical-geometry-g12-maths-t4p2","status":"publish","type":"post","link":"https:\/\/repstaq.com\/study-guide-docs\/analytical-geometry-g12-maths-t4p2\/","title":{"rendered":"Analytical-Geometry-G12-Maths-T4P2"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">Analytical Geometry Study Notes<\/h2>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Distance between Two Points<\/h2>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The distance between two points on the Cartesian plane measures the straight-line length connecting them. This formula comes from the Pythagorean Theorem.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Important Facts:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>You must know the coordinates of both points.<\/li>\n\n\n\n<li>The order of subtraction does not affect the result because the terms are squared.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Formula:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nd = sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\n<\/span><\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate the distance between A(2 ; 3) and B(7 ; 11).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 1: Identify the coordinates.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nA(2,3),quad B(7,11)\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 2: Write the formula.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nd = sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 3: Substitute the values.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nd = sqrt{(7-2)^2 + (11-3)^2}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 4: Simplify.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n= sqrt{5^2 + 8^2}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n= sqrt{25 + 64}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n= sqrt{89}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 5: Final answer.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nd = sqrt{89}\n<\/span><\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Remember:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Always write final answers in simplest surd form unless asked for a decimal.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Common Mistake:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Swapping coordinates but not squaring may lead to negative or incorrect results.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/sqooltutors.co.za\/sign-up-b\/\"><img decoding=\"async\" src=\"https:\/\/lmxddlwowqlsyucleuyv.supabase.co\/storage\/v1\/object\/public\/pdf_ads\/batch%201\/Doc%20Image%20Aug%2022,%202026,%2004_21_14%20PM%20(1).png\" alt=\"\"\/><\/a><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Gradient of a Line<\/h2>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The gradient of a line describes its steepness and direction. It is calculated as the ratio of the vertical change to the horizontal change between two points.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Important Facts:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The gradient is denoted by m.<\/li>\n\n\n\n<li>Parallel lines have equal gradients.<\/li>\n\n\n\n<li>A positive gradient means the line rises as it moves from left to right; a negative gradient means it falls.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Formula:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nm = frac{y_2 - y_1}{x_2 - x_1}\n<\/span><\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Find the gradient of the line passing through Q(\u20134 ; \u20136) and R(12 ; 2).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 1: Identify the coordinates.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nQ(-4,-6),quad R(12,2)\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 2: Write the formula.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nm = frac{y_2 - y_1}{x_2 - x_1}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 3: Substitute the values.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nm = frac{2 - (-6)}{12 - (-4)}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 4: Simplify.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n= frac{2 + 6}{12 + 4}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n= frac{8}{16}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n= frac{1}{2}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 5: Final answer.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nm = frac{1}{2}\n<\/span><\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Exam Tip:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Always check for division by zero (vertical lines have undefined gradients).<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/sqooltutors.co.za\/sign-up-b\/\"><img decoding=\"async\" src=\"https:\/\/lmxddlwowqlsyucleuyv.supabase.co\/storage\/v1\/object\/public\/pdf_ads\/batch%201\/Doc%20Image%20Aug%2022,%202026,%2004_21_26%20PM%20(1).png\" alt=\"\"\/><\/a><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Equation of a Straight Line<\/h2>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The equation of a straight line relates x and y so that all (x, y) pairs on the line satisfy the equation.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Important Facts:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The standard form of the equation is y = mx + c, where m is the gradient and c is the y-intercept.<\/li>\n\n\n\n<li>The y-intercept is found where x=0.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Formula:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">y = mx + c<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Find the equation of the line that passes through Q(\u20134 ; \u20136) with m = \u00bd.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 1: Identify given values.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nQ(-4, -6),quad m = frac{1}{2}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 2: Write the formula.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\ny = mx + c\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 3: Substitute the gradient and the point.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n-6 = frac{1}{2}(-4) + c\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 4: Simplify.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n-6 = -2 + c\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nc = -6 + 2\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nc = -4\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 5: Write the final equation.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\ny = frac{1}{2}x - 4\n<\/span><\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Common Mistake:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Mixing up x- and y-coordinates when substituting into the equation.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Angle of Inclination<\/h2>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The angle of inclination ( \u03b8 ) of a line is the angle measured counterclockwise from the positive x-axis to the line.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Important Facts:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>0 degrees means the line is parallel to the x-axis (horizontal).<\/li>\n\n\n\n<li>Angles between 0 and 180 degrees.<\/li>\n\n\n\n<li>The angle of inclination is linked to the gradient: m = tan \u03b8.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Formula:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\ntan \u03b8 = m\n<\/span><\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Find the angle of inclination of a line with m = \u00bd.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 1: Write the formula.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\ntan \u03b8 = m\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 2: Substitute the value for m.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\ntan \u03b8 = frac{1}{2}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 3: Calculate \u03b8.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n\u03b8 = arctan(frac{1}{2})\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n\u03b8 \u2248 26,57\u00b0 \n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 4: State the final answer.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nAngle of inclination = 26,57\u00b0\n<\/span><\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Exam Tip:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Use your calculator in degree mode for all CAPS examination calculations.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Angle between Two Lines<\/h2>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The angle between two straight lines measures how far one line turns to coincide with the other.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Important Facts:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The formula involves the gradients of both lines.<\/li>\n\n\n\n<li>If both gradients are known, the acute angle \u03b8 between them is found by:<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Formula:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\ntan \u03b8 = frac{|m_1 - m_2|}{1 + m_1 m_2}\n<\/span><\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Find the angle between two lines with m\u2081 = 2 and m\u2082 = \u20131.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 1: Write the formula.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\ntan \u03b8 = frac{|m_1 - m_2|}{1 + m_1 m_2}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 2: Substitute the values.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\ntan \u03b8 = frac{|2 - (-1)|}{1 + (2)(-1)}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 3: Simplify.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n= frac{|2 + 1|}{1 - 2}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n= frac{3}{-1}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n= -3\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 4: Calculate \u03b8.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n\u03b8 = arctan(3)\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n\u03b8 \u2248 71,57\u00b0\n<\/span><\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/sqooltutors.co.za\/sign-up-b\/\"><img decoding=\"async\" src=\"https:\/\/lmxddlwowqlsyucleuyv.supabase.co\/storage\/v1\/object\/public\/pdf_ads\/batch%201\/Doc%20Image%20Aug%2022,%202026,%2004_21_31%20PM%20(1).png\" alt=\"\"\/><\/a><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Parallel and Perpendicular Lines<\/h2>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Parallel lines have the same gradient, while perpendicular lines have gradients that multiply to \u20131.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Important Facts:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Parallel: m\u2081 = m\u2082<\/li>\n\n\n\n<li>Perpendicular: m\u2081 \u00d7 m\u2082 = \u20131<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">If line 1 has gradient m\u2081 = 3, what is the gradient of a line perpendicular to it?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 1: Write the relationship for perpendicular gradients.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nm_1 \u00d7 m_2 = \u20131\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 2: Substitute the value for m\u2081.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n3 \u00d7 m_2 = \u20131\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 3: Solve for m\u2082.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nm_2 = \u2013frac{1}{3}\n<\/span><\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Common Mistake:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Forgetting to take the negative reciprocal for perpendicular lines.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Application:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Used to show lines in geometric figures are parallel or perpendicular, such as in rectangles or proving rhombus properties.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Midpoint Theorem<\/h2>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The midpoint of a line segment divides it into two equal parts. The coordinates of the midpoint are the averages of the corresponding coordinates of the endpoints.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Formula:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">For points P( x_1 ; y_1 ) and Q( x_2 ; y_2 ),<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nM(x_m; y_m) = left( frac{x_1 + x_2}{2}; frac{y_1 + y_2}{2} right)\n<\/span><\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Find the midpoint between P(\u20131 ; 8) and Q(\u20134 ; \u20136).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 1: Identify the coordinates.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nP(-1, 8),quad Q(-4, -6)\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 2: Write the formula.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nM = left( frac{-1 + (-4)}{2}, frac{8 + (-6)}{2} right)\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 3: Simplify.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n= left( frac{-5}{2}, frac{2}{2} right)\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n= left( -2,5;\\ 1 right)\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 4: State the answer.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nMidpoint = (-2,5 ; 1)\n<\/span><\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Exam Tip:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Double check arithmetic when adding or dividing negative numbers.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Coordinate Geometry Proofs<\/h2>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Coordinate geometry proofs use algebraic methods with coordinates, gradients, distances and midpoints to prove geometric properties of figures on the Cartesian plane.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Common Proof Types:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Prove a quadrilateral is a parallelogram by showing both pairs of opposite sides are parallel or equal.<\/li>\n\n\n\n<li>Prove a triangle is isosceles by showing two sides are equal using the distance formula.<\/li>\n\n\n\n<li>Prove a point is the midpoint of a segment using the midpoint formula.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Show that triangle PQR with P(\u20131 ; 8), Q(\u20134 ; \u20136), and R(12 ; 2) is isosceles.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 1: Find PQ.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nPQ = sqrt{(\u20134 \u2013 (\u20131))^2 + (\u20136 \u2013 8)^2}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n= sqrt{(\u20133)^2 + (\u201314)^2}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n= sqrt{9 + 196}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n= sqrt{205}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 2: Find PR.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nPR = sqrt{(12 \u2013 (\u20131))^2 + (2 \u2013 8)^2}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n= sqrt{13^2 + (\u20136)^2}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n= sqrt{169 + 36}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n= sqrt{205}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 3: Statement.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since <span class=\"katex-eq\" data-katex-display=\"false\">PQ = PR<\/span>, triangle PQR is isosceles.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Exam Tip:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>State clearly what you are proving and reference your calculation.<\/li>\n\n\n\n<li>Use correct mathematical notation for all steps.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Conclusion<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Analytical Geometry in Grade 12 covers essential tools for exploring and proving properties of geometric figures using algebraic methods. Mastery of these techniques is important for examinations. Always show your working clearly, use step by step reasoning, and double-check calculations for accuracy.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Analytical Geometry Study Notes Distance between Two Points Explanation: The distance between two points on the Cartesian plane measures the straight-line length connecting them. This formula comes from the Pythagorean Theorem. Important Facts: Formula: Example: Calculate the distance between A(2 ; 3) and B(7 ; 11). Step 1: Identify the coordinates. Step 2: Write the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_breakdance_hide_in_design_set":false,"_breakdance_tags":"","footnotes":""},"categories":[3],"tags":[8],"class_list":["post-38","post","type-post","status-publish","format-standard","hentry","category-g12","tag-pure-maths"],"acf":{"document_name":"Analytical Geometry","grade":12,"subject":"Mathematics","term":4,"paper":2},"_links":{"self":[{"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/posts\/38","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/comments?post=38"}],"version-history":[{"count":25,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/posts\/38\/revisions"}],"predecessor-version":[{"id":219,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/posts\/38\/revisions\/219"}],"wp:attachment":[{"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/media?parent=38"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/categories?post=38"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/tags?post=38"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}