Question

If A and B are points \((-6, 7)\) and \((-1, -5)\), then three times the length of AB is equal to ?

• A. 169
• B. 13
• C. 39
• D. 26

Hint:

1. Use the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Points are A\((-6, 7)\) and B\((-1, -5)\). 2. Calculate differences: \(x_2 - x_1 = -1 - (-6) = -1 + 6 = 5\) \(y_2 - y_1 = -5 - 7 = -12\) 3. Square the differences: \((5)^2 = 25\) \((-12)^2 = 144\) 4. Sum the squares and take the square root: \(AB = \sqrt{25 + 144} = \sqrt{169} = 13\). 5. Three times the length: \(3 \times 13 = 39\).

Answer 1

The Correct Option is C

Concept: The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a Cartesian coordinate system is given by the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Step 1: Identify the coordinates of points A and B Point A = \((x_1, y_1) = (-6, 7)\) Point B = \((x_2, y_2) = (-1, -5)\) Step 2: Calculate the length of AB using the distance formula \[ AB = \sqrt{((-1) - (-6))^2 + ((-5) - 7)^2} \] \[ AB = \sqrt{(-1 + 6)^2 + (-5 - 7)^2} \] \[ AB = \sqrt{(5)^2 + (-12)^2} \] \[ AB = \sqrt{25 + 144} \] \[ AB = \sqrt{169} \] \[ AB = 13 \] The length of AB is 13 units. Step 3: Calculate three times the length of AB We need to find \(3 \times AB\). \[ 3 \times AB = 3 \times 13 \] \[ 3 \times AB = 39 \] Three times the length of AB is 39. This matches option (3).