Gr12 Mathematics: June Easy P(2)

<p>Quadrilateral</p> Signup and view all the answers

<p>$\frac{1}{2} \times diagonal AC \times diagonal BD$</p> Signup and view all the answers

<p>The corresponding sides are in proportion and the corresponding angles are equal</p> Signup and view all the answers

<p>$\frac{1}{2} \times (base_1 + base_2) \times height$</p> Signup and view all the answers

<p>(x - a)^2 + (y - b)^2 = r^2</p> Signup and view all the answers

<p>Pythagoras' Theorem</p> Signup and view all the answers

<p>$\frac{1}{2} \times diagonal AC \times diagonal BD$</p> Signup and view all the answers

<p>m_radius * m_tangent = -1</p> Signup and view all the answers

<p>The corresponding sides are in proportion and the corresponding angles are equal</p> Signup and view all the answers

<p>Cross Multiplication</p> Signup and view all the answers

<p>Thales' Theorem</p> Signup and view all the answers

<p>A straight line that touches the circle at exactly one point</p> Signup and view all the answers

<p>To put the equation of the circle in standard form</p> Signup and view all the answers

<p>The radius is perpendicular to the tangent at the point of tangency</p> Signup and view all the answers

<p>m_tangent = -1 / m_radius</p> Signup and view all the answers

<p>A comparison of two quantities with the same units</p> Signup and view all the answers

<p>To compare the quantities more easily</p> Signup and view all the answers

<p>Area = 1/2 × base × height</p> Signup and view all the answers

<p>Their areas are proportional to their bases.</p> Signup and view all the answers

<p>A line drawn parallel to one side of a triangle divides the other two sides proportionally.</p> Signup and view all the answers

<p>The line joining the midpoints of two sides of a triangle is parallel to the third side and half its length.</p> Signup and view all the answers

<p>They have the same shape but differ in size and orientation.</p> Signup and view all the answers

<p>All pairs of corresponding angles are equal, and all pairs of corresponding sides are in the same proportion.</p> Signup and view all the answers

<p>The line drawn from the midpoint of one side of a triangle parallel to another side bisects the third side.</p> Signup and view all the answers

<p>Area1 : Area2 = Base1 : Base2</p> Signup and view all the answers

<p>They have equal areas.</p> Signup and view all the answers

<p>They are similar and have proportional areas.</p> Signup and view all the answers

<p>Both b and c</p> Signup and view all the answers

<p>If two triangles have equal corresponding angles, they are similar.</p> Signup and view all the answers

<p>$\frac{AB}{DE} = \frac{AC}{DF} = \frac{BC}{EF}$</p> Signup and view all the answers

<p>Proportionality theorem</p> Signup and view all the answers

<p>The two triangles are similar.</p> Signup and view all the answers

<p>Pythagorean theorem</p> Signup and view all the answers

<p>Right-angled triangle with a square drawn on each side</p> Signup and view all the answers

<p>Area = 0.5 * base * height</p> Signup and view all the answers

<p>Triangles with equal heights have areas proportional to their bases.</p> Signup and view all the answers

<p>Corresponding sides are in proportion.</p> Signup and view all the answers

<p>x^2 + y^2 = 25</p> Signup and view all the answers

<p>line y = -x + 1</p> Signup and view all the answers

<p>center: (-2, 3), radius = 4</p> Signup and view all the answers

<p>Divide both sides of the equation by the coefficient of x^2 and y^2</p> Signup and view all the answers

<p>(x - 3)^2 + (y + 2)^2 = 49</p> Signup and view all the answers

<p>x^2 + y^2 = 25</p> Signup and view all the answers

<p>(x + 1)^2 + (y - 2)^2 = 9</p> Signup and view all the answers

<p>√13</p> Signup and view all the answers

<p>θ = sin^(-1) x + k ⋅ 360°</p> Signup and view all the answers

<p>Area = (1/2)bc sin A</p> Signup and view all the answers

<p>When no right angle is given, and three sides are given</p> Signup and view all the answers

<p>h = (d sin α) / (sin β) ⋅ tan β</p> Signup and view all the answers

<p>When no perpendicular height is given</p> Signup and view all the answers

<p>θ = tan^(-1) x + k ⋅ 180°</p> Signup and view all the answers

<p>Area = √(s(s-a)(s-b)(s-c))</p> Signup and view all the answers

<p>When no right angle is given, and two sides and an angle (not the included angle) are given</p> Signup and view all the answers

<p>h = (b sin α sin θ) / (sin(β + θ))</p> Signup and view all the answers

<p>Draw a sketch, consider the given information, apply appropriate rules, and calculate the desired quantities</p> Signup and view all the answers

<p>They are similar.</p> Signup and view all the answers

<p>The Unit Circle Definition</p> Signup and view all the answers

<p>cos(α - β) = cos α cos β + sin α sin β</p> Signup and view all the answers

<p>The Law of Cosines</p> Signup and view all the answers

<p>To establish proportions between corresponding sides.</p> Signup and view all the answers

<p>They are complementary angles.</p> Signup and view all the answers

<p>Two sides of one triangle are congruent to two sides of the other triangle.</p> Signup and view all the answers

<p>The Negative Angle Identity</p> Signup and view all the answers

<p>Using the area of similar triangles to derive a relationship between the sides.</p> Signup and view all the answers

<p>sin(α + β) = sin α cos β + cos α sin β and sin(α - β) = sin α cos β - cos α sin β</p> Signup and view all the answers

<p>$ an(eta - eta)$</p> Signup and view all the answers

<p>$ an(eta + eta)$</p> Signup and view all the answers

<p>$rac{ an 2 heta}{2}$</p> Signup and view all the answers

<p>Deriving from the sine of a sum</p> Signup and view all the answers

<p>$$rac{ an^2}{ an^2}$$</p> Signup and view all the answers

<p>$2 an eta$</p> Signup and view all the answers

<p>Using a CAST diagram and reference angles</p> Signup and view all the answers

<p>$ an(eta - eta)$</p> Signup and view all the answers

<p>$eta = heta$</p> Signup and view all the answers

<p>x^2 + y^2 = r^2</p> Signup and view all the answers

<p>(x - a)^2 + (y - b)^2 = r^2</p> Signup and view all the answers

<p>It is symmetric about the x-axis, y-axis, and origin.</p> Signup and view all the answers

<p>To find the center and radius of a circle.</p> Signup and view all the answers

<p>(x - 3)^2 + (y + 2)^2 = 49</p> Signup and view all the answers

<p>The x-axis, y-axis, origin, and lines y = x and y = -x.</p> Signup and view all the answers

<p>Center (-2, 3) and radius 4.</p> Signup and view all the answers

<p>The line y = 2x.</p> Signup and view all the answers

<p>To determine the equation of a circle in standard form</p> Signup and view all the answers

<p>Their product is -1</p> Signup and view all the answers

<p>A comparison of two quantities with the same units</p> Signup and view all the answers

<p>Cross Multiplication</p> Signup and view all the answers

<p>Thales' theorem</p> Signup and view all the answers

<p>(x - a)^2 + (y - b)^2 = r^2</p> Signup and view all the answers

<p>A straight line that touches the circle at exactly one point</p> Signup and view all the answers

<p>-1 / m_radius</p> Signup and view all the answers

<p>They have proportional sides</p> Signup and view all the answers

<p>To find the equation of a tangent to a circle</p> Signup and view all the answers

<p>If they have the same height.</p> Signup and view all the answers

<p>The ratio of their bases.</p> Signup and view all the answers

<p>They have equal areas.</p> Signup and view all the answers

<p>If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.</p> Signup and view all the answers

<p>The line joining the midpoints of two sides of a triangle is parallel to the third side and half its length.</p> Signup and view all the answers

<p>If a line is drawn from the midpoint of one side of a triangle parallel to another side, it bisects the third side.</p> Signup and view all the answers

<p>Polygons with the same shape but different sizes.</p> Signup and view all the answers

<p>All pairs of corresponding angles are equal and all pairs of corresponding sides are in the same proportion.</p> Signup and view all the answers

<p>They are similar.</p> Signup and view all the answers

<p>Their corresponding sides are in proportion.</p> Signup and view all the answers

<p>All pairs of corresponding sides are in the same proportion</p> Signup and view all the answers

<p>The corresponding sides are in the same proportion</p> Signup and view all the answers

<p>The square on the hypotenuse is equal to the sum of the squares on the other two sides</p> Signup and view all the answers

<p>Triangles with equal heights have areas proportional to their bases</p> Signup and view all the answers

<p>Construction of a triangle</p> Signup and view all the answers

<p>Their areas are proportional to their bases</p> Signup and view all the answers

<p>If the corresponding sides of two triangles are in proportion, then the two triangles are similar</p> Signup and view all the answers

<p>They are similar</p> Signup and view all the answers

<p>Equiangular triangles theorem</p> Signup and view all the answers

<p>All pairs of corresponding angles are equal or all pairs of corresponding sides are in the same proportion</p> Signup and view all the answers

<p>A plane, closed shape consisting of three or more line segments.</p> Signup and view all the answers

<p>Area = rac{1}{2} imes (base1 + base2) imes height</p> Signup and view all the answers

<p>Equiangular triangle</p> Signup and view all the answers

<p>Their corresponding angles are equal, and corresponding sides are in proportion.</p> Signup and view all the answers

<p>Basic Proportionality Theorem (Thales' Theorem)</p> Signup and view all the answers

<p>Its diagonals are perpendicular.</p> Signup and view all the answers

<p>Ellipse</p> Signup and view all the answers

<p>Area = rac{1}{2} imes diagonal AC imes diagonal BD</p> Signup and view all the answers

<p>The hypotenuse is the longest side.</p> Signup and view all the answers

<p>Any one of the sides of the triangle.</p> Signup and view all the answers

<p>(\angle A_1 = \angle C), (\angle B = \angle A_2), (\angle D_1 = \angle C)</p> Signup and view all the answers

<p>(\angle A_2 = \angle B), (\angle C = \angle A_1), (\angle D_2 = \angle C)</p> Signup and view all the answers

<p>(rac{AB}{BC} = rac{BD}{AB})</p> Signup and view all the answers

<p>(rac{AC}{CB} = rac{DC}{AC})</p> Signup and view all the answers

<p>(BC^2 = AB^2 + AC^2)</p> Signup and view all the answers

<p>If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the angle included by these two sides is a right angle.</p> Signup and view all the answers

<p>(rac{1}{2} \cdot b \cdot h)</p> Signup and view all the answers

<p>All corresponding angles are equal and all corresponding sides are in the same proportion.</p> Signup and view all the answers

<p>(\cos(\alpha - eta) = \cos \alpha \cos eta + \sin \alpha \sin eta)</p> Signup and view all the answers

<p>(\sin(\alpha + eta) = \sin \alpha \cos eta + \cos \alpha \sin eta)</p> Signup and view all the answers

<p>(\cos^2 \alpha - \sin^2 \alpha)</p> Signup and view all the answers

<p>Use the sum formula for sine: (\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta)</p> Signup and view all the answers

<p>(\sin^2 \alpha - \cos^2 \alpha)</p> Signup and view all the answers

<p>The sum formula for cosine: (\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta)</p> Signup and view all the answers

<p>(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta)</p> Signup and view all the answers

<p>(\frac{24}{25})</p> Signup and view all the answers

<p>Use the Pythagorean Identity: (\sin^2 \alpha + \cos^2 \alpha = 1) to simplify the expression further</p> Signup and view all the answers

<p>(\frac{7}{9})</p> Signup and view all the answers

<p>(\frac{2\tan \alpha}{1 - \tan^2 \alpha})</p> Signup and view all the answers

<p>(\frac{4}{5})</p> Signup and view all the answers

<p>(\theta = \tan^{-1} x + k \cdot 180^\circ)</p> Signup and view all the answers

<p>Cosine Rule</p> Signup and view all the answers

<p>(\text{Area} = \frac{1}{2}bc\sin A)</p> Signup and view all the answers

<p>When no right angle is present, and two sides along with a non-included angle are given</p> Signup and view all the answers

<p>(h = \frac{d \sin \alpha}{\sin \beta})</p> Signup and view all the answers

<p>(\frac{1}{2}bc\sin A)</p> Signup and view all the answers

<p>(\theta = \sin^{-1} x + k \cdot 360^\circ) or (\theta = 180^\circ - \sin^{-1} x + k \cdot 360^\circ)</p> Signup and view all the answers

<p>Three sides of the triangle are given</p> Signup and view all the answers

<p>(\frac{AB}{\sin C} = \frac{BC}{\sin A})</p> Signup and view all the answers

<p>x^2 + y^2 = r^2</p> Signup and view all the answers

<p>(x - a)^2 + (y - b)^2 = r^2</p> Signup and view all the answers

<p>The x-axis, the y-axis, the origin, and the lines y = x and y = -x</p> Signup and view all the answers

<p>To find the center and radius of the circle</p> Signup and view all the answers

<p>(x - a)^2 + (y - b)^2 = r^2</p> Signup and view all the answers

<p>5</p> Signup and view all the answers

<p>(-2, 3)</p> Signup and view all the answers

<p>(x + 1)^2 + (y - 2)^2 = 9</p> Signup and view all the answers

<p>Their areas are proportional to their bases.</p> Signup and view all the answers

<p>The line joining the midpoints of two sides of a triangle is parallel to the third side and half its length.</p> Signup and view all the answers

<p>Their corresponding sides are proportional.</p> Signup and view all the answers

<p>Their areas are equal.</p> Signup and view all the answers

<p>A line drawn parallel to one side of a triangle divides the other two sides proportionally.</p> Signup and view all the answers

<p>They have the same shape but differ in size.</p> Signup and view all the answers

<p>All pairs of corresponding angles are equal and all pairs of corresponding sides are in the same proportion.</p> Signup and view all the answers

<p>The line drawn from the midpoint of one side of a triangle parallel to another side bisects the third side of the triangle.</p> Signup and view all the answers

<p>They are similar.</p> Signup and view all the answers

<p>Area1 / Area2 = base1 / base2</p> Signup and view all the answers

<p>$\frac{1}{2} \times \text{base} \times \text{height}$</p> Signup and view all the answers

<p>Corresponding angles are equal.</p> Signup and view all the answers

<p>Basic Proportionality Theorem</p> Signup and view all the answers

<p>$\text{base} \times \text{height}$</p> Signup and view all the answers

<p>Diagonals are equal.</p> Signup and view all the answers

<p>$\frac{1}{2} \times (\text{base}_1 + \text{base}_2) \times \text{height}$</p> Signup and view all the answers

<p>Corresponding angles are congruent.</p> Signup and view all the answers

<p>Pythagorean Theorem</p> Signup and view all the answers

<p>$\frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2$</p> Signup and view all the answers

<p>Quadrilateral</p> Signup and view all the answers

<p>(\theta = 180^\circ - \sin^{-1} x + k \cdot 360^\circ)</p> Signup and view all the answers

<p>When two sides and an angle (not the included angle) are given.</p> Signup and view all the answers

<p>Area Rule</p> Signup and view all the answers

<p>(h = \frac{d \sin \alpha}{\sin \beta} \tan \beta)</p> Signup and view all the answers

<p>Draw a sketch</p> Signup and view all the answers

<p>Two angles and a side are given.</p> Signup and view all the answers

<p>All of the above.</p> Signup and view all the answers

<p>(\theta = \cos^{-1} x + k \cdot 360^\circ)</p> Signup and view all the answers

<p>Cosine Rule</p> Signup and view all the answers

<p>(\theta = \tan^{-1} x + k \cdot 180^\circ)</p> Signup and view all the answers

<p>To simplify the equation of a circle in general form</p> Signup and view all the answers

<p>Their product is -1</p> Signup and view all the answers

<p>A relationship between two quantities with the same units</p> Signup and view all the answers

<p>Cross multiplication</p> Signup and view all the answers

<p>Thales' theorem</p> Signup and view all the answers

<p>(x - a)^2 + (y - b)^2 = r^2</p> Signup and view all the answers

<p>A straight line that touches the circle at exactly one point</p> Signup and view all the answers

<p>To make them easier to compare</p> Signup and view all the answers

<p>x^2 + y^2 = 25</p> Signup and view all the answers

<p>They are similar</p> Signup and view all the answers

<p>All pairs of corresponding angles are equal, or all pairs of corresponding sides are in proportion.</p> Signup and view all the answers

<p>The triangles are similar.</p> Signup and view all the answers

<p>The corresponding angles of the triangles are equal.</p> Signup and view all the answers

<p>Establishing that a line parallel to one side creates corresponding angles.</p> Signup and view all the answers

<p>All sides must be congruent.</p> Signup and view all the answers

<p>The sides being proportional.</p> Signup and view all the answers

<p>It relates the lengths of sides of right-angled triangles.</p> Signup and view all the answers

<p>The length of side BC squared equals the sum of the squares of sides AB and AC.</p> Signup and view all the answers

<p>They have equal areas.</p> Signup and view all the answers

<p>Drawing parallels to establish angle connections.</p> Signup and view all the answers

<p>$egin{align*} ext{sin} heta ext{cos} eta - ext{cos} heta ext{sin} eta ewline ext{or } ext{sin} ( heta - eta) ext{ } ext{if } heta,eta= ext{c}. ext{} ext{} ext{} ext{ } heta,eta ext{ angles} ext{ } ext{ using complex angle formulas. } ext{ } ext{} ext{ } ext{} ext{ } ext{ } ext{ } ext{ } ext{ } ext{ angles} ext{ } ext{ } ext{ and arcs are cancels.} ext{ } ext{ } ext{ } ext{} ext{ } ext{ } ext{ } ext{ } ext{ } ewline ext{origin are many such complex numbers included.} ext{ } ext{ } ext{ } ewline ext{} ext{} ext{ } ewline ext{ angles } ext{ } ext{ and their hyperbolic counterparts.} ext{ } ext{ } ext{ } ewline} ext{ angle.} ext{ }$</p> Signup and view all the answers

<p>$ ext{cos} heta ext{cos} eta - ext{sin} heta ext{sin} eta$</p> Signup and view all the answers

<p>$2 ext{sin} heta ext{cos} heta$</p> Signup and view all the answers

<p>$ ext{cos}^2 heta - ext{sin}^2 heta$</p> Signup and view all the answers

<p>Simplify the equation using algebraic methods</p> Signup and view all the answers

<p>Where the function is positive or negative</p> Signup and view all the answers

<p>$ ext{sin} heta ext{cos} eta + ext{cos} heta ext{sin} eta$</p> Signup and view all the answers

<p>$ ext{cos}( heta - eta) = ext{cos} heta ext{cos} eta + ext{sin} heta ext{sin} eta$</p> Signup and view all the answers

<p>Finding the derivative of the function</p> Signup and view all the answers