In every rectangle, the two diagonals have the same length. If a quadrilateral's diagonals have the same length, that doesn't mean it has to be a rectangle, but if a parallelogram's diagonals have the same length, then it's definitely a rectangle.

So first of all, let's prove this is a parallelogram. The basic definition of a parallelogram is that it is a quadrilateral where both pairs of opposite sides are parallel.

So let's name the vertices as:

$A(-1,3) \\ \\ B(1,5) \\ \\ C(5,1) \\ \\ D(3,-1)$

First pair of opposite sides:

<u>Slope:</u>

$\text{For AB}: \\ \\ m=\frac{5-3}{1-(-1)}=1 \\ \\ \\ \text{For CD}: \\ \\ m=\frac{1-(-1)}{5-3}=1 \\ \\ \\ \text{So AB and CD are parallel}$

Second pair of opposite sides:

<u>Slope:</u>

$\text{For BC}: \\ \\ m=\frac{1-5}{5-1}=-1 \\ \\ \\ \text{For AD}: \\ \\ m=\frac{-1-3}{3-(-1)}=-1 \\ \\ \\ \text{So BC and AD are parallel}$

So in fact this is a parallelogram. The other thing we need to prove is that the diagonals measure the same. Using distance formula:

$d=\sqrt{(y_{2}-y_{1})^2+(x_{2}-x_{1})^2} \\ \\ \\ Diagonal \ BD: \\ \\ d=\sqrt{(5-(-1))^2+(1-3)^2}=2\sqrt{10} \\ \\ \\ Diagonal \ AC: \\ \\ d=\sqrt{(3-1)^2+(-5-1)^2}=2\sqrt{10} \\ \\ \\$

So the diagonals measure the same, therefore this is a rectangle.

5 0

3 years ago

I need a little extra help with number three

diamong [38]

Answer:

The answer to your question is:

Step-by-step explanation:

Data

f(x) = -2x² + 8x - 2

Process

-2x² + 8x = y + 2

-2(x² - 4x + 4) = y + 2 - 8

-2(x - 2)² = y - 6

(x - 2)² = 1/2 (y - 6)

Vertex = (2, 6)

Axis of symmetry = x = 2

y-intercept

f(0) = -2(0)² + 8(0) - 2

f(0) = 0 + 0 - 2

f(0) = -2

Domain (-∞, ∞)

Range (-∞, 6]

See the graph below

6 0

3 years ago

What is the equation of the line?

suter [353]

Answer:

y=-7

Step-by-step explanation:

when it is hohorizontal lile this it is y=-7

7 0

3 years ago