Understand Slope

lessonbunny.com

Answer Key & Explanations

Understand Slope · Grade 8 · Worksheet 3

1. A line passes through the points (3, 5) and (7, 13). What is the slope of this line? Answer: 2 Solution: To find the slope of a line passing through two points, we use the slope formula: slope = (change in y) / (change in x) = (y2 - y1) / (x2 - x1) Identify the coordinates of the two points.
   Full step-by-step solution

   To find the slope of a line passing through two points, we use the slope formula: slope = (change in y) / (change in x) = (y2 - y1) / (x2 - x1) Step 1: Identify the coordinates of the two points. Point 1: (3, 5) so x1 = 3 and y1 = 5 Point 2: (7, 13) so x2 = 7 and y2 = 13 Step 2: Substitute these values into the slope formula. slope = (y2 - y1) / (x2 - x1) slope = (13 - 5) / (7 - 3) Step 3: Calculate the difference in the y-values (the rise). 13 - 5 = 8 Step 4: Calculate the difference in the x-values (the run). 7 - 3 = 4 Step 5: Divide the rise by the run to get the slope. slope = 8 / 4 Step 6: Simplify the fraction. 8 / 4 = 2 Therefore, the slope of the line is 2.

2. A line is drawn on a coordinate plane that passes through points (3, 7) and (9, 19). Another line is drawn perpendicular to this line and passes through point (6, 13). What is the slope of this perpendicular line? Answer: -1/2 Solution: Find the slope of the first line using the formula (y2 - y1)/(x2 - x1) Using points (3, 7) and (9, 19): slope = (19 - 7)/(9 - 3) = 12/6 = 2 For perpendicular lines, the slopes are negative reciprocals of each other The negative reciprocal of 2 is -1/2 Therefore, the slope of the perpendicular…
   Full step-by-step solution

   Step 1: Find the slope of the first line using the formula (y2 - y1)/(x2 - x1) Step 2: Using points (3, 7) and (9, 19): slope = (19 - 7)/(9 - 3) = 12/6 = 2 Step 3: For perpendicular lines, the slopes are negative reciprocals of each other Step 4: The negative reciprocal of 2 is -1/2 Step 5: Therefore, the slope of the perpendicular line is -1/2

3. (-2 - 4) ÷ (3 - (-1)) = ? Answer: -1.5 Solution: Simplify the numerator: -2 - 4 = -6 Simplify the denominator: 3 - (-1) = 3 + 1 = 4 Divide the results: -6 ÷ 4 = -1.5 The answer is -1.5.
   Full step-by-step solution

   Step 1: Simplify the numerator: -2 - 4 = -6 Step 2: Simplify the denominator: 3 - (-1) = 3 + 1 = 4 Step 3: Divide the results: -6 ÷ 4 = -1.5 The answer is -1.5.

4. (2x - 8)/4 = 3 Answer: 10 Solution: Multiply both sides by 4 to eliminate the fraction: (2x - 8)/4 × 4 = 3 × 4 This simplifies to: 2x - 8 = 12 Add 8 to both sides: 2x - 8 + 8 = 12 + 8 This simplifies to: 2x = 20 Divide both sides by 2: 2x/2 = 20/2 This gives: x = 10 The answer is 10.
   Full step-by-step solution

   Step 1: Multiply both sides by 4 to eliminate the fraction: (2x - 8)/4 × 4 = 3 × 4 Step 2: This simplifies to: 2x - 8 = 12 Step 3: Add 8 to both sides: 2x - 8 + 8 = 12 + 8 Step 4: This simplifies to: 2x = 20 Step 5: Divide both sides by 2: 2x/2 = 20/2 Step 6: This gives: x = 10 The answer is 10.

5. Ava is watching a hot air balloon rise. The balloon rises 33 meters while moving horizontally 11 meters. What is the slope of the balloon's path? Answer: 3 Solution: Identify the rise and run. The vertical rise is 33 meters, and the horizontal run is 11 meters. Use the slope formula: slope = rise / run.
   Full step-by-step solution

   Step 1: Identify the rise and run. The vertical rise is 33 meters, and the horizontal run is 11 meters. Step 2: Use the slope formula: slope = rise / run. Step 3: Substitute the values: slope = 33 / 11. Step 4: Simplify: 33 / 11 = 3. The slope of the balloon's path is 3.

6. A line passes through the points (3, 7) and (5, 13). What is the slope of this line? Answer: 3 Solution: To find the slope of a line that passes through two points, we use the slope formula: slope = (change in y) / (change in x) = (y2 - y1) / (x2 - x1) Identify the coordinates of the two points.
   Full step-by-step solution

   To find the slope of a line that passes through two points, we use the slope formula: slope = (change in y) / (change in x) = (y2 - y1) / (x2 - x1) Step 1: Identify the coordinates of the two points. Point 1: (x1, y1) = (3, 7) Point 2: (x2, y2) = (5, 13) Step 2: Substitute the coordinates into the slope formula. slope = (13 - 7) / (5 - 3) Step 3: Perform the subtractions inside the parentheses. slope = (6) / (2) Step 4: Divide the numerator by the denominator. slope = 6 / 2 = 3 Therefore, the slope of the line is 3.

7. A line is drawn on a coordinate plane passing through points A(-2, 4) and B(4, -2). A second line is drawn perpendicular to this line and passes through point C(1, 5). What is the slope of the second line? Answer: 1 Solution: Find the slope of the first line using points A(-2, 4) and B(4, -2) Slope = (y2 - y1)/(x2 - x1) = (-2 - 4)/(4 - (-2)) = (-6)/(6) = -1 Find the slope of a line perpendicular to the first line Perpendicular slope = negative reciprocal of -1 = 1 The second line has slope 1 Point C(1, 5) is not…
   Full step-by-step solution

   Step 1: Find the slope of the first line using points A(-2, 4) and B(4, -2) Slope = (y2 - y1)/(x2 - x1) = (-2 - 4)/(4 - (-2)) = (-6)/(6) = -1 Step 2: Find the slope of a line perpendicular to the first line Perpendicular slope = negative reciprocal of -1 = 1 Step 3: The second line has slope 1 Point C(1, 5) is not needed to find the slope of the perpendicular line The answer is 1.

8. A right triangle is drawn on a coordinate plane with vertices at (1, 1), (4, 1), and (4, 5). What is the slope of the hypotenuse? Answer: 1.3333333333333333 Solution: Identify the vertices of the triangle. We have points: (1, 1), (4, 1), and (4, 5). Determine which two points form the hypotenuse.
   Full step-by-step solution

   Let's find the slope of the hypotenuse step by step. Step 1: Identify the vertices of the triangle. We have points: (1, 1), (4, 1), and (4, 5). Step 2: Determine which two points form the hypotenuse. In a right triangle, the hypotenuse is the side opposite the right angle. Check the points: - Between (1, 1) and (4, 1): y-coordinates are the same (y = 1), so this is a horizontal line. - Between (4, 1) and (4, 5): x-coordinates are the same (x = 4), so this is a vertical line. These two sides are perpendicular, so the right angle is at (4, 1). Thus, the hypotenuse is between (1, 1) and (4, 5). Step 3: Recall the slope formula. Slope m = (y2 - y1) / (x2 - x1) Step 4: Assign coordinates. Let (x1, y1) = (1, 1) and (x2, y2) = (4, 5). Step 5: Substitute into the formula. m = (5 - 1) / (4 - 1) m = 4 / 3 Step 6: Convert to decimal form. 4 divided by 3 = 1.3333333333333333 Final Answer: 1.3333333333333333

9. A right triangle is drawn on a coordinate plane with vertices at (2, 1), (8, 1), and (8, 5). What is the slope of the hypotenuse? Answer: 2/3 Solution: Identify the vertices of the right triangle. The points are (2, 1), (8, 1), and (8, 5). Determine which side is the hypotenuse.
   Full step-by-step solution

   Let's find the slope of the hypotenuse step-by-step. Step 1: Identify the vertices of the right triangle. The points are (2, 1), (8, 1), and (8, 5). Step 2: Determine which side is the hypotenuse. The hypotenuse is the side opposite the right angle. Check the coordinates: - (2, 1) and (8, 1) have the same y-coordinate (y = 1), so this is a horizontal line. - (8, 1) and (8, 5) have the same x-coordinate (x = 8), so this is a vertical line. These two sides are perpendicular, so the right angle is at (8, 1). Thus, the hypotenuse is the side between (2, 1) and (8, 5). Step 3: Recall the slope formula. Slope = (change in y) / (change in x) = (y2 - y1) / (x2 - x1) Step 4: Apply the slope formula to the hypotenuse endpoints. Let (x1, y1) = (2, 1) and (x2, y2) = (8, 5). Change in y = y2 - y1 = 5 - 1 = 4 Change in x = x2 - x1 = 8 - 2 = 6 Step 5: Calculate the slope. Slope = 4 / 6 Step 6: Simplify the fraction. 4/6 = 2/3 Final Answer: The slope of the hypotenuse is 2/3.