Parallelogram abcd has vertices at a(1,2) , b(0,9) , c(7,8) , and d(8,1) . which conclusion can be made?

1. ac⊥bd ; therefore, abcd is a rhombus.

2. ac=bd ; therefore, abcd is a rhombus.

3. ac⊥bd ; therefore, abcd is a rectangle.

4. ac=bd ; therefore, abcd is a rectangle.

Solution:

Vertices of parallelogram ABCD is given as A(1,2) , B(0,9) , C(7,8) , and D(8,1) .

We will use the following analytical geometry formulas here

1. Distance between two points

2. Slope of a line

3. When two lines are perpendicular, product of their slopes is equal to -1.

Since ABCD is a parallelogram

1. Opposite sides are equal and parallel.

2. Diagonals bisect each other.

3. Opposite angles are equal.

Now, coming to problem

As, you can see that, AB=BC=CD=DA=√ 50

But , slope of AB × Slope of BC =slope of CB × Slope of DC=slope of CD × Slope of DA

which is not equal to -1. It means lines which are sides of parallelogram are not perpendicular.

As, all side of parallelogram ABCD are equal, so it is a rhombus.

As, diagonal of rhombus bisect each other at right angles.

Shows that diagonals are perpendicular bisector of each other.

Option (1) : AC⊥BD; therefore, ABCD is a rhombus.

1. AC ⊥BD ; therefore, ABCD is a rhombus.

Step-by-step explanation:

We are given,

A parallelogram ABCD with vertices A(1,2), B(0,9), C(7,8) and D(8,1).

Using Distance Formula, which gives the distance between two points ( and ( as

So, the distance between two points are given by,

AB= = 7.07

BC= = 7.07

CD= = 7.07

DA= = 7.07

Also,

AC= = 8.49

BD= = 11.3

Since, we have,

AB = BC = CD = DA.

So, ABCD is a rhombus but not a rectangle.

As, AC ≠ BD. Moreover, two diagonals are perpendicular to each other in a rhombus. Thus, AC ⊥ BD.

Hence, we get that, Option A is correct.

step-by-step explanation:

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