Question

the distance between the points (-4, -3) and (5, y) is 9√2. use the distance formula to find a possible value of y. y = □

Explanation:

Step1: Recall Distance Formula

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Here, \((x_1, y_1)=(-4, -3)\), \((x_2, y_2)=(5, y)\), and \(d = 9\sqrt{2}\). Substitute these into the formula:

$$9\sqrt{2}=\sqrt{(5 - (-4))^2 + (y - (-3))^2}$$

Step2: Simplify Inside the Square Root

Simplify \(5 - (-4)=9\) and \(y - (-3)=y + 3\). So the equation becomes:

$$9\sqrt{2}=\sqrt{9^2 + (y + 3)^2}$$

Step3: Square Both Sides

Square both sides to eliminate the square root:

$$(9\sqrt{2})^2 = 9^2 + (y + 3)^2$$

Calculate \((9\sqrt{2})^2 = 81\times2 = 162\) and \(9^2 = 81\). Then:

$$162 = 81 + (y + 3)^2$$

Step4: Solve for \((y + 3)^2\)

Subtract 81 from both sides:

$$(y + 3)^2 = 162 - 81 = 81$$

Step5: Take Square Root and Solve for y

Take the square root of both sides: \(y + 3=\pm\sqrt{81}=\pm9\)
Case 1: \(y + 3 = 9\) → \(y = 9 - 3 = 6\)
Case 2: \(y + 3 = -9\) → \(y = -9 - 3 = -12\) (either 6 or -12 is a valid solution)

Answer:

6 (or -12, here we choose 6 as a possible value)