Proofs with Variable Coordinates

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Proofs with Variable Coordinates
Page 13: #’s 17-21
17.The vertices of quadrilateral RSTV are R(0,0),
S(a,0), T(a+b,c) and V(b,c)
a) Find the slopes of RV and ST
RV
m
y2  y1
x2  x1
c0
m
b0
c
m
b
ST
c0
m
( a  b)  a
m
c
b
17.The vertices of quadrilateral RSTV are R(0,0),
S(a,0), T(a+b,c) and V(b,c)
b) Find the lengths of RV and ST
ST
RV
d
d
x2  x1 2   y2  y1 2
b  0
2
 c  0
d  b2  c 2
2
d
x2  x1 2   y2  y1 2
d
a  b  a 2  c  02
d  b2  c 2
c) Since one pair of opposite sides has equal slopes, they
are parallel. The same pair of opposite sides are equal in
length. Quadrilateral RSTV with one pair of opposite sides
both parallel and congruent is a parallelogram.
18. The vertices of triangle ABC are A(0,0), B(4a,0),
C(2a,2b)
a) Find the coordinates of D, the midpoint of AC.
AC
x1  x2
y1  y 2
xm 
ym 
2
2
0  2a
0  2b
xm 
ym 
2
2
2a
2b
xm 
a
ym 
b
2
2
M AC  D(a, b)
18. The vertices of triangle ABC are A(0,0), B(4a,0),
C(2a,2b)
b) Find the coordinates of E, the midpoint of BC.
BC
x1  x2
y1  y 2
xm 
ym 
2
2
4a  2a
0  2b
xm 
ym 
2
2
6a
2b
xm 
 3a
ym 
b
2
2
M BC  E(3a, b)
18. The vertices of triangle ABC are A(0,0), B(4a,0),
C(2a,2b)
E (3a, b)
c) Show that AB=2DE
DE
AB
d
d
x2  x1 2   y2  y1 2
d
4a  02  0  02
d
d  4a
2
x2  x1 2   y2  y1 2
3a  a 2  b  b 2
d  ( 2a ) 2  0 2
d  ( 4a ) 2  0 2
d  16a
D(a, b)
AB  2 DE
4a ? 2(2a)
4a  4a
d  4a2
d  2a
19. The vertices of quadrilateral ABCD are A(0,0), B(a,0),
C(a,b) and D(0,b)
a) Show that ABCD is a parallelogram
AB
BC
CD
y2  y1
m
x2  x1
b0
m
aa
bb
m
a0
00
m
a0
b
m
0
0
m 0
a
0
m 0
a
undefined
DA
b0
m
00
b
m
0
undefined
Since the slopes of both pairs of opposite sides are equal,
they are parallel. Therefore quadrilateral ABCD with both
pair of opposite sides parallel is a parallelogram.
19. The vertices of quadrilateral ABCD are A(0,0), B(a,0),
C(a,b) and D(0,b)
b) Show that diagonal AC is congruent to diagonal BD
AC
d
d
BD
x2  x1 2   y2  y1 2
a  0
2
 b  0 
d  a 2  b2
2
x2  x1 2   y2  y1 2
d
a  02  0  b 2
2
d  a 2   b 
d
d  a 2  b2
C) The diagonals of parallelogram ABCD are congruent. A
parallelogram with congruent diagonals is a rectangle.
20. The vertices of quadrilateral ABCD are A(0,0), B(r,s),
C(r,s+t) and D(0,t)
a) Represent the slopes of AB and CD
AB
CD
y2  y1
m
x2  x1
s t t
m
r 0
s0
m
r 0
s
m
r
s
m
r
Since the slopes of the opposite sides are equal,
they are parallel.
20. The vertices of quadrilateral ABCD are A(0,0), B(r,s),
C(r,s+t) and D(0,t)
b) Represent the lengths of AB and CD
CD
AB
d
d
x2  x1 2   y2  y1 2
d
x2  x1 2   y2  y1 2
r  02  s  02
d
r  02  s  t  t 2
d  r 2  s2
d  r 2  s2
AB  CD
C) Since quadrilateral ABCD has the same pair of opposite
sides (AB and CD) both parallel and congruent, it is a
parallelogram.
21. The vertices of triangle RST are R(0,0), S(2a,2b), T(4a,0)
The midpoints of RS, ST, TR are L, M, and N, respectively.
a) Express the coordinates of the midpoints in terms of a and b.
ST
RS
y  y2
x x
xm  1 2 y m  1
2
2
xm 
TR
y  y2
x1  x2
ym  1
2
2
xm 
y  y2
x1  x2
ym  1
2
2
0  2b
ym 
2
xm 
0  2b
2a  4a
ym 
2
2
xm 
0  4a
2
2b
2a
b
xm 
 a ym 
2
2
xm 
2b
6a
b
 3a ym 
2
2
xm 
0
4a
 2a ym   0
2
2
0  2a
xm 
2
La, b 
M 3a, b 
ym 
N 2a,0
00
2
21. The vertices of triangle RST are R(0,0), S(2a,2b), T(4a,0)
The midpoints of RS, ST, TR are L, M, and N, respectively.
b) ProveLM RT
La, b 
M 3a, b 
RT
LM
00
4a  0
y2  y1
m
x2  x1
m
bb
m
3a  a
0
m
0
4a
m
0
0
2a
Since the slopes of LM and RT are equal, they are parallel.
21. The vertices of triangle RST are R(0,0), S(2a,2b), T(4a,0)
The midpoints of RS, ST, TR are L, M, and N, respectively.
c) ProveSN  RT
N 2a,0
SN
m
y2  y1
x2  x1
RT
m0
2b  0
m
2a  2a
2b
m
0
undefined
Since SN has an undefined slope, it is a vertical line. RT
has a zero slope so it is a horizontal line. Therefore SN is
perpendicular to RT.
21. The vertices of triangle RST are R(0,0), S(2a,2b), T(4a,0)
The midpoints of RS, ST, TR are L, M, and N, respectively.
d) ProveRST is isosceles
d
d
x2  x1 2   y2  y1 2
d
x2  x1 2   y2  y1 2
2a  02  2b  02
d
4a  2a 2  0  2b 2
d  ( 2a ) 2  ( 2b) 2
d  4a 2  4b2
TR
ST
RS
d
d
x2  x1 2   y2  y1 2
4a  02  0  02
d  (2a ) 2  (2b) 2
d  ( 4a ) 2  0 2
d  4a 2  4b2
d  16a2
d  4a
Since two sides of the triangle are congruent, triangle RST
is isosceles.

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