This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Welcome back lost — been a while! Let's pick up where you left off. Question 11: To find the distance between two points (x_1, y_1) and (x_2, y_2), we use the distance formula: D = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2) Step 1: Identify the coordinates. Given points are (-2, 2) and (1, 6). So, x_1 = -2, y_1 = 2, x_2 = 1, y_2 = 6. Step 2: Substitute the values into the distance formula. D = sqrt((1 - (-2))^2 + (6 - 2)^2) D = sqrt((1 + 2)^2 + (4)^2) D = sqrt((3)^2 + (4)^2) Step 3: Calculate the squares and sum them. D = sqrt(9 + 16) D = sqrt(25) Step 4: Find the square root. D = 5 The distance is 5 units. The correct answer is b) 5 units. --- Question 12: Angles that sum up to 90^ are defined as complementary angles. The correct answer is a) complementary angles. --- Question 13: An infinite number of lines can pass through a single point. Imagine drawing lines through the origin (0,0) at every possible angle; there are infinitely many such lines. The correct answer is d) **. --- Question 14: To find the midpoint of two points (x_1, y_1) and (x_2, y_2), we use the midpoint formula: M = ((x_1 + x_2)/(2), (y_1 + y_2)/(2)) Step 1: Identify the coordinates. Given points are (-2, 8) and (8, -2). So, x_1 = -2, y_1 = 8, x_2 = 8, y_2 = -2. Step 2: Substitute the values into the midpoint formula. M = ((-2 + 8)/(2), (8 + (-2))/(2)) Step 3: Calculate the sums and divide by 2. M = ((6)/(2), (6)/(2)) M = (3, 3) The coordinates of the midpoint are (3, 3). The correct answer is b) (3,3). --- Question 15: Given the equation x + (1)/(x) = 2. We need to find the value of x^2 + (1)/(x^2). Step 1: Square both sides of the given equation. (x + (1)/(x))^2 = 2^2 Step 2: Expand the left side using the formula (a+b)^2 = a^2 + 2ab + b^2. x^2 + 2 · x · (1)/(x) + ((1)/(x))^2 = 4 Step 3: Simplify the equation. x^2 + 2 + (1)/(x^2) = 4 Step 4: Isolate x^2 + (1)/(x^2). x^2 + (1)/(x^2) = 4 - 2 x^2 + (1)/(x^2) = 2 The value of x^2 + (1)/(x^2) is 2. The correct answer is d) 2. Drop the next question! 📸