Laura is designing a rectangular porch for her house. On her scale drawing, the vertices of the rectangle are (2, 7), (8, 7), and (2, 9). What are the coordinates of the fourth vertex?
A. (8, 9)
B. (7, 9)
C. (7, 2)
D. (2, 8)
We can use the fact that opposite sides of a rectangle are parallel and congruent to find the fourth vertex. The side of the rectangle connecting (2,7) and (8,7) is the same length as the side connecting the fourth vertex and (2,9). Therefore, the x-coordinate of the fourth vertex must be the same as that of (8,7), which is 8. Similarly, the side connecting (2,7) and (2,9) is congruent to the side connecting the fourth vertex and (8,7), so the y-coordinate of the fourth vertex must be the same as that of (2,9), which is 9. Therefore, the fourth vertex is (8,9), which is answer choice (A).
Which ordered pair is not a solution of y = –12x – 3?
A. (–3, 33)
B. (8, –98)
C. (–9, 105)
D. (3, –39)
To check if an ordered pair is a solution of the equation $y=-12x-3$, we substitute the values of $x$ and $y$ into the equation and see if it is a true statement.
For choice (A), we have $y=-12(-3)-3=33$, so the ordered pair $(x,y)=(-3,33)$ is a solution.
For choice (B), we have $y=-12(8)-3=-99$, so the ordered pair $(x,y)=(8,-98)$ is not a solution.
For choice (C), we have $y=-12(-9)-3=105$, so the ordered pair $(x,y)=(-9,105)$ is a solution.
For choice (D), we have $y=-12(3)-3=-39$, so the ordered pair $(x,y)=(3,-39)$ is a solution.
Therefore, the only pair that is not a solution is $(x,y)=\boxed{\text{(B) }(8,-98)}$.
Find three solutions of the equation.
y = 2x – 6
A. (–2, –10), (1, –4), (2, –1)
B. (–2, –10), (1, –4), (0, –6)
C. (0, –6), (3, –2), (–2, –10)
D. (1, –4), (0, –6), (–1, –9)
We can find solutions by choosing values for $x$ and then computing the corresponding value for $y$ using the equation $y=2x-6$.
For choice (A), plugging in $x=-2$, $x=1$, and $x=2$ gives us the values $y=-10$, $y=-4$, and $y=-2$, respectively. However, none of these values match the $y$-coordinate of any of the answer choices.
For choice (B), plugging in $x=-2$, $x=1$, and $x=0$ gives us the values $y=-10$, $y=-4$, and $y=-6$, respectively. These solutions match the $y$-coordinates in answer choice (B).
For choice (C), plugging in $x=0$, $x=3$, and $x=-2$ gives us the values $y=-6$, $y=0$, and $y=-10$, respectively. These solutions match the $y$-coordinates in answer choice (C).
For choice (D), plugging in $x=1$, $x=0$, and $x=-1$ gives us the values $y=-4$, $y=-6$, and $y=-8$, respectively. None of these values match the $y$-coordinate of any of the answer choices.
Therefore, the three solutions to the equation $y=2x-6$ are $\boxed{\text{(B) }( -2, -10 )}$, $\boxed{\text{(B) }( 1, -4 )}$, and $\boxed{\text{(B) }( 0, -6 )}$.
To find the coordinates of the fourth vertex of the rectangular porch, we can use the fact that opposite sides of a rectangle are equal in length and parallel to each other.
In this case, we have two sides of the rectangle: one side is defined by the vertices (2, 7) and (8, 7), and the other side is defined by the vertices (2, 7) and (2, 9).
First, let's calculate the length of the side defined by (2, 7) and (8, 7). We can subtract the x-coordinates:
8 - 2 = 6
So, the length of this side is 6.
Now, let's calculate the length of the side defined by (2, 7) and (2, 9). We can subtract the y-coordinates:
9 - 7 = 2
So, the length of this side is 2.
Since opposite sides of a rectangle are equal in length, we know that the length of the side defined by (2, 7) and (8, 7) is the same as the length of the side defined by (2, 7) and the fourth vertex.
Therefore, the x-coordinate of the fourth vertex must be 8.
Similarly, since opposite sides of a rectangle are equal in length, we know that the length of the side defined by (2, 7) and (2, 9) is the same as the length of the side defined by (8, 7) and the fourth vertex.
Therefore, the y-coordinate of the fourth vertex must be 9.
So, the coordinates of the fourth vertex are (8, 9).
Therefore, the correct answer is A. (8, 9).