Question

ira is making a clock and wants to make sure that all of the numbers line up across from each other perfectly. using the cartesian plane, she marks the place where the hands will be attached at (-3, -7), in the centre of the clock. next she marks where she will place the one, at (9, 19), as shown in the diagram below. where should ira mark the place for the seven, directly across from the one?

Explanation:

Step1: Identify center and point

Center \( C = (-3, -7) \), point for 1 \( P = (9, 19) \).

Step2: Find midpoint formula

Let point for 7 be \( Q = (x, y) \). Center is midpoint: \( \frac{9+x}{2} = -3 \), \( \frac{19+y}{2} = -7 \).

Step3: Solve for x

\( 9 + x = -6 \implies x = -15 \)? No, wait: \( 2*(-3) = 9 + x \implies x = -6 -9 = -15 \)? Correction: Wait, midpoint formula is \( (\frac{x1+x2}{2}, \frac{y1+y2}{2}) = center \). So \( \frac{9 + x}{2} = -3 \implies 9 + x = -6 \implies x = -15 \); \( \frac{19 + y}{2} = -7 \implies 19 + y = -14 \implies y = -33 \). Yes, correct.

Step4: Final coordinates

\( Q = (-15, -33) \)? Wait no: Wait \( 2(-3) = 9 + x \implies x = -6 -9 = -15 \); \( 2(-7) =19 + y \implies y= -14 -19 = -33 \). Yes, that's right.

Wait correction: The center is (-3,-7). The point for 1 is (9,19). The point for 7 is directly opposite, so center is midpoint between 1 and7. So:

Let (x,y) be the point for7. Then:

\( ( (-3), (-7) ) = ( \frac{9 + x}{2}, \frac{19 + y}{2} ) \)

So solving for x: \( 9 + x = 2*(-3) = -6 \implies x = -6 -9 = -15 \)

Solving for y: \(19 + y =2*(-7)= -14 \implies y= -14 -19 = -33 \)

So the point is (-15,-33).

Step1: Recall midpoint property

Center is midpoint of opposite points.

Step2: Set up midpoint equations

Let (x,y) = 7's coordinates.
\( \frac{9+x}{2}=-3 \), \( \frac{19+y}{2}=-7 \)

Step3: Solve for x

\(9+x=-6 \implies x=-15\)

Step4: Solve for y

\(19+y=-14 \implies y=-33\)

Answer:

(-21, -33)