Question

Geometry problem help!

Please refer to the image below...

Answer 1

A. 1/3

B. √10

C. -1, 1

D. √8, 6

E. congruent and opposite pairs parallel

F. perpendicular, not congruent

G. rhombus, explanation below

Explanation:

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A.

Slope is the rise over the run. Let's look at F to G.

We are going from -1 to 2 on our x-axis (run), so our run is 3 units.

Our rise is 1 unit as we go from 2 to 3 on the y-axis.

`slope=(rise)/(run) =(1)/(3)`

This slope is the same for all of the sides.

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B.

We will use the distance formula (which is basically just the Pythagorean Theorem) to calculate the length of each side. Let's go between F and G again, but this distance is the same for all the sides.

`\sqrt{(x_(2)-x_1)^2+(y_(2)-y_1)^2 } \\\\(x_1,y_1)=(-1,2)\\\\(x_2,y_2)=(2,3)\\\\\\√((2+1)^2+(3-2)^2 ) \\\\√((3)^2+(1)^2 )\\\\√(9+1 )\\\\√(10)`

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C.

The diagonals are the lines that connect the non-adjacent vertices.

Our two diagonals are FH and GE.

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FH

We go from x-value -1 to 1 from F to H, so our run is 2.

We go from y-value 2 to 0. so our rise is -2.

`slope=(rise)/(run) =-(2)/(2) =-1`

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GE

We go from x-value -2 to 2 from E to G, so our run is 4.

We go from y-value -1 to 3. so our rise is 4.

`slope=(rise)/(run) =(4)/(4) =1`

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D.

Let's use the distance formula on each of our diagonals.

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FH

`\sqrt{(x_(2)-x_1)^2+(y_(2)-y_1)^2 } \\\\(x_1,y_1)=(-1,2)\\\\(x_2,y_2)=(1,0)\\\\\\√((1+1)^2+(0-2)^2 ) \\\\√((2)^2+(-2)^2 )\\\\√(4+4 )\\\\√(8)`

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GE

`\sqrt{(x_(2)-x_1)^2+(y_(2)-y_1)^2 } \\\\(x_1,y_1)=(-2,-1)\\\\(x_2,y_2)=(2,3)\\\\\\√((2+2)^2+(3+1)^2 ) \\\\√((4)^2+(4)^2 )\\\\√(16+16 )\\\\√(36)\\\\6`

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E.

They are congruent as they all have the same length (√10) and the opposite sides are parallel as they have the same slope (1/3)

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F.

They are perpendicular diagonals as their slopes are negative reciprocals (1 and -1), and they are not congruent as they have different lengths (√8 and 6).

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G.

Parallelogram- quadrilateral with opposite pairs of parallel sides.

Rhombus- a parallelogram with four equal sides

Square- a rhombus with four right angles

We can see that this is a parallelogram as we saw that the opposite sides are parallel due to having the same slope, and the perpendicular diagonals show that as well. This is also a rhombus because if we use that distance formula on all the sides, it will be the same. It is not a square though because it does not have four right angles, so this is a rhombus.

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