Three vertices of a rectangle have coordinates (3,4), (5-4), and (-7-7) What is the #y# coordinate of the fourth vertex?

1 Answer
May 20, 2018

#" "#
#color(blue)("y-coordinate value of the fourth vertex [ D ] = 1"#

Explanation:

#" "#
#color(green)("Step 1 :"#

Plot the given points in sequence with Vertex A, Vertex B and Vertex C respectively.

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Joint the points #A, B and C# with line segments #AB and BC# as shown.

#color(green)("Step 2 :"#

Use the distance formula to find the magnitudes of the line segments #AB and BC#.

This process explains how it is done using Algebra:

Distance Formula:

#color(blue)(d=sqrt[(x_2-x_1)^2 + (y_2-y_1)^2]#

(a) Distance between the points : #A(3,4) and B(5,-4)#

#x_1=3, y_1=4, x_2=5 and y_2=-4#

#d=sqrt[(5-3)^2+{(-4)-4}^2#

#d=sqrt(2^2+(-8)^2#

#d=sqrt(68)#

#d~~ 8.246211251#

#d=8.25# (2 decimal places)

Hence, the line segment #AB = 8.25# units.

(b) Distance between the points : #B(5,-4 ) and c(-7,-7 )#

#x_1=5, y_1=-4, x_2=-7 and y_2=-7#

#d=sqrt[{(-7)-5}^2+{(-7)-(-4)}^2#

#d=sqrt(144+(-3)^2#

#d=sqrt(144+9) = sqrt(153)~~12.36931688#

#d=12.37# (2 decimal places)

Hence, the line segment #BC = 12.37# units.

#color(green)("Step 3 :"#

Verify these results with geometrical constructions:

(a) Draw a parallel line through the point #A#, with the line parallel to #BC#

(b) Draw a parallel line through the point #C#, with the line parallel to #AB#

(c) #D# is the point of intersection of lines.

(d) Connect points #A and D# and the points #C and D#.

We have a parallelogram.

Measure the lengths of the line segments #AB, BC, CD and AD#.

We find that #AB=8.25, BC=12.37, CD=8.25, AD=12.37#

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#AB||CD# and they are also congruent.

#AD||BC# and they are also congruent.

We have two pairs of opposite sides that are parallel and equal.

There are 4 right angles.

Hence, #ABCD# is a rectangle.

Vertex of D to be #(-9, 1)#

Hence the y-coordinate of the Vertex D is #1#.