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 = \square

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, \(x_1=-4\), \(y_1 = - 3\), \(x_2 = 5\), \(y_2=y\), and \(d = 9\sqrt{2}\). Substitute these values 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:

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

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:

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

Step3: Solve for \((y + 3)^2\)

Subtract 81 from both sides:

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

Step4: Solve for \(y\)

Take the square root of both sides: \(y + 3=\pm\sqrt{81}=\pm9\)
Case 1: \(y + 3 = 9\) then \(y=9 - 3 = 6\)
Case 2: \(y + 3=-9\) then \(y=-9 - 3=-12\)

Answer:

A possible value of \(y\) is \(6\) (or \(-12\))