proofs with variable coordinates page 13: #’s 17-21
TRANSCRIPT
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
12
12
xx
yym
RV
0
0
b
cm
b
cm
ST
aba
cm
)(
0
b
cm
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
RV
212
212 yyxxd
22 00 cbd
22 cbd
ST
212
212 yyxxd
22 0 cabad
22 cbd
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.
221 xx
xm
2
20 axm
aa
xm 2
2
AC
221 yy
ym
2
20 bym
bb
ym 2
2
),( baDM AC
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.
221 xx
xm
2
24 aaxm
aa
xm 32
6
BC
221 yy
ym
2
20 bym
bb
ym 2
2
),3( baEM BC
18. The vertices of triangle ABC are A(0,0), B(4a,0), C(2a,2b)
c) Show that AB=2DE
AB
212
212 yyxxd
22 0004 ad
22 0)4( ad
216ad
ad 4
DE
212
212 yyxxd
223 bbaad
22 0)2( ad
24ad
ad 2
),3( baE ),( baD
DEAB 2)2(24 aa
aa 44
?
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
12
12
xx
yym
AB
0
00
a
m
00
a
m
BC
aa
bm
0
0
bm
undefined
CD
0
a
bbm
00
a
m
DA
00
0
b
m
0
bm
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 BD
C) The diagonals of parallelogram ABCD are congruent. A parallelogram with congruent diagonals is a rectangle.
212
212 yyxxd
22 00 bad
22 bad
212
212 yyxxd
22 00 bad
22 bad
22 bad
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
12
12
xx
yym
AB
0
0
r
sm
r
sm
CD
0
r
ttsm
r
sm
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
AB CD
C) Since quadrilateral ABCD has the same pair of opposite sides (AB and CD) both parallel and congruent, it is a parallelogram.
212
212 yyxxd
22 00 srd
22 srd
212
212 yyxxd
220 ttsrd
22 srd
CDAB
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.
221 xx
xm
2
20 axm
aa
xm 2
2
RS
221 yy
ym
2
20 bym
bb
ym 2
2
221 xx
xm
2
42 aaxm
aa
xm 32
6
ST
221 yy
ym
2
20 bym
bb
ym 2
2
221 xx
xm
2
40 axm
aa
xm 22
4
TR
221 yy
ym
2
00 my
02
0my
baL , baM ,3 0,2aN
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.
baL ,
baM ,3
RTLM Prove b)
12
12
xx
yym
LM
aa
bbm
3
02
0
am
RT
04
00
a
m
04
0
am
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.
RTSN Prove c)
12
12
xx
yym
SN
aa
bm
22
02
0
2bm
RT
0m 0,2aN
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.
isosceles is Prove d) RST
Since two sides of the triangle are congruent, triangle RST is isosceles.
RS ST
212
212 yyxxd
22 0202 bad
22 )2()2( bad
212
212 yyxxd
22 2024 baad
22 )2()2( bad
TR
212
212 yyxxd
22 0004 ad
22 0)4( ad
22 44 bad 22 44 bad 216ad
ad 4