Analytic geometry #1 - Questions and Answers

Question 1 What shapes are quadrilateral? Answer 1 Kite, diamond, rhombus, square, rectangle, parallelogram, trapezoid are quadrilateral. Answer 2 Shapes which are all quadrilaterals: square, rectangle, rhombus, trapezium, parallelogram and kite. Answer 3 A quadrilateral is a polygon that has exactly four sides. (This also means that a quadrilateral has exactly four vertices, and exactly four angles) They are: rectangle, rhombus, square, trapezium, kite and parallelogram. Question 2 Find the area of the shaded region. r² = sin(2θ) Answer A = ∫(1/2) r² dθ = ∫(1/2) sin(2θ) dθ = (1/4) ∫ sin(2θ) dθ = -1/4 [ cos(2θ) ] from 0 to π/2 = -1/4 [ cos(2*π/2) - cos(0) ] = -1/4 [ -1 - 1 ] = 1/2 Question 3 Find the area of the parallelogram with vertices A(-3, 0), B(-1, 3), C(5, 2), and D(3, -1). Answer 1 We can choose any three of the supplied points and find a two vector that expresses the sides of the parallelogram "or one side and a diagonal of the parallelogram," we can choose the points B, C, and D. Furthermore, we can choose point B as the common point for the two sides "the initial point of the two vectors," so we must find vector BD and vector BC using the equation below. BC = <5+1, 2-3> = <6,-1> And, the other vector side BD is BD = <3+1,-1-3> = <4,-4> Furthermore, the cross product of both vectors is BC x BD = <6,-1,0> x <4,-4,0> = | i j k | | 6 -1 0 | | 4 -4 0 | = i |(-1 * 0) - (0 * -4)| - j |(6 * 0) - (0 * 4)| + k |(6 * -4) - (-1 * 4)| = 0 i + 0 j + k (-24 + 4) = -20k Knowing the cross product of the parallelogram's two vectors, we may apply an equation to get the area. Area = |-20k| = 20 As a result, the parallelogram's area is 20 units squared. Answer 2 Step 1: Given: Area = 1/2 |(x₂-x₁)(y₃-y₁) - (x₄-x₁)(y₂-y₁)| Let's calculate the area using the given vertices A(-3, 0), B(-1, 3), C(5, 2), and D(3, -1). We can label the coordinates as follows: A(-3,0), B(-1, 3), C(5, 2), and D(3, - 1). Step 2: Now, we substitute these values into the formula: Area = 1/2 |(-1-(-3))(2-0)-(3-(-3))(3-0)| Simplifying further: Area = 1/2 |(2)(2) - (6)(3)| Area = 1/2 |4-18| Area = 1/2 |-14| Taking the absolute value: Area = 14/2 Finally, evaluating the expression: Area = 7 Therefore, the area of the parallelogram with vertices A(-3, 0), B(-1, 3), C(5, 2), and D(3, -1) is 7 square units. Answer 3 Area = 1/2 |(x1-x3) (y2 - y4) - (x2 - x4) (y1 - y3)| Substituting the coordinates of the given vertices, we have: Area = 1/2 |(( - 3) - 5)(3 – ( − 1)) – (( − 1) − 3)(0-2))| Simplifying the expression: Area = 1/2 |((-8)(4) - (-4)(-2))| Area = 1/2 |(-32-8)| Area = 1/2 |-40| Area = 1/2 * 40 = 20 Hence, the area of the parallelogram is 20 square units. Answer 4 To find the area of the parallelogram with vertices A(-3, 0), B(-1, 3), C(5, 2), and D(3, -1), we can use the formula: Area = 1/2 |x1y2 + x2y3 + x3y4 + x4y1 - x2y1 - x3y2 - x4y3 - x1y4| where (x1, y1), (x2, y2), (x3, y3), (x4, y4) are the coordinates of the four vertices of the parallelogram. Substituting the given coordinates into the formula, we have: Area = 1/2 |((-3)(3) + (-1)(2) + (5)(-1) + (3)(0) - (-1)(0) - (5)(3) - (3)(-1) - (-3)(2))| Simplifying the expression inside the absolute value: Area = 1/2 |(-9 - 2 - 5 + 0 + 0 - 15 + 3 + 6)| Area = 1/2 |-22| Area = |-11| Therefore, the area of the parallelogram is 11 square units. Question 4 The measure of an angle is 6 less than 5 times its complement. What is the measure of the complement? Answer 1 Two angles are complementary if the sum of their measures is 90°. If x is the measure of the angle, then 90-x is the measure of the complement, both in degrees. Using the given statement, we can write: x = 5(90-x) - 6 Solve for x: x = 450 - 5x - 6 x = 444 - 5x 6x = 444 x = 74 So, the measure of the complement is 90 - 74 = 16°. Answer 2 Step 1: Let's assume the measure of the angle is x. According to the problem statement, the measure of an angle is 6 less than 5 times its complement. The complement of an angle is the angle that, when added to the given angle, results in a right angle (90 degrees). The complement of x can be represented as 90 - x. Given that the measure of the angle is 6 less than 5 times its complement, we can write the equation as: x = 5(90-x) - 6 To solve this equation, let's simplify it: x = 450 - 5x - 6 Step 2: Now, let's gather like terms: x + 5x = 450 - 6 6x = 444 To isolate x, we divide both sides of the equation by 6: 6x/6 = 444/6 x = 74 Therefore, the measure of the angle is 74 degrees. Step 3: To find the measure of the complement, we substitute x back into the expression for the complement: Complement = 90 - x Complement = 90 - 74 = 16 Hence, the measure of the complement is 16 degrees. To summarize, the measure of the angle is 74 degrees, and the measure of its complement is 16 degrees. Answer 3 According to the given information, we have the equation: x = 5y - 6 To find the measure of the complement, we need to solve for y. Rearranging the equation, we get: 5y = x + 6 Dividing both sides by 5, we have: y= (x + 6) / 5 Therefore, the measure of the complement is (x+6) / 5 Answer 4 According to the problem, we can set up the following equation: x = 5(90° - x) - 6 To solve this equation, we'll start by distributing the 5: x = 450° - 5x - 6 Next, we'll combine like terms: x + 5x = 450° - 6 6x = 444° Finally, we'll solve for x by dividing both sides of the equation by 6: x = 444° / 6 x = 74° Therefore, the measure of the angle is 74°. To find the measure of its complement, we substitute the value of x back into the equation for the complement: Complement = 90° - x Complement = 90° - 74° Complement = 16° Hence, the measure of the complement is 16°. Question 5 Consider the line represented by the equation 5x + 2y = 10. How is the slope of the line related to values of A, B, and C in standard form Ax + By = C? Answer Consider the standard form of equation: Ax + By = C Isolate the variable y term on one side of the equation: By = -Ax + C Divide both sides of the equation with B to write the equation in slope intercept form: y = (-A / B)x + C/B This implies that the slope of an equation written in standard form is -A/B therefore the slope of 5x + 2y = 10 is -5/2