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 Solution 1 (a) 

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This time we begin by reflecting over the line

 Translate 

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 Reflect 

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 Undo Translation 

We will begin by reflecting △PQR over the line

 Solution 2 (a) 

 Translate 

 Undo Translation 

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 Back to Solutions 

△PQR  is congruent to △ABC. We will see multiple ways that 
show this using a sequence of translations and reflections. 

P

Q

R

A

B

C

P =

Q

R

R =

Q =

First we translate △PQR so that P is the same point as A

Now we reflect △PQR over the line 

$l$

$l$.

△PQR is congruent to △ABC!

$l$

Q

R

P

P

R

Q

$l$.

We will now translate △PQR so that Q is the same as A.

= Q

= P

= R

We can now see that △PQR is congruent to △ABC.

We will now translate △PQR so that P equals A

Q

P

R

P

Q

R

We can now see that △PQR is congruent to △ABC

= P

= Q

= R

$l$.

$l$

$l$

P

Q

R

We will begin by translating △PQR down by the vector shown.

P

Q

R

We will no reflect △PQR over the line

$l$.

P

Q

R

We will now reflect △PQR over the line

$l$.

= P

= R

= Q

Finally, we can see that △PQR is congruent to △ABC.

A = (-7, 5)
B = (-1, 4)
C = (4, 9)
poly1, c, a, b = Polygon(A, B, C)
D = (6, 5)
u = Vector(C, D)
A' = Translate(A, u)
B' = Translate(B, u)
C' = Translate(C, u)
poly1', c', a', b' = Polygon(A', B', C')
E = Point(u)
v = Vector(C, E)
A'_1 = Translate(A, v)
B'_1 = Translate(B, v)
C'_1 = Translate(C, v)
poly1'_1, c'_1, a'_1, b'_1 = Polygon(A'_1, B'_1, C'_1)
F = (6, 3)
d = Line(D, F)
A'' = Mirror(A', d)
B'' = Mirror(B', d)
C'' = Mirror(C', d)
poly1'', c'', a'', b'' = Polygon(A'', B'', C'')
G = Mirror(A'_1, d)
H = Mirror(B'_1, d)
I = Mirror(C'_1, d)
poly2, i, g, h = Polygon(G, H, I)
A'_2 = Mirror(A, d)
B'_2 = Mirror(B, d)
C'_2 = Mirror(C, d)
poly1'_2, c'_2, a'_2, b'_2 = Polygon(A'_2, B'_2, C'_2)
J = (5, 9)
e = Line(J, d)
K = (5, 5)
w = Vector(J, K)
L = Mirror(A'_1, e)
M = Mirror(B'_1, e)
N = Mirror(C'_1, e)
O = Point(w)
z = Vector(J, O)
f = (0.019999999999999574, 0.019999999999999574)
L' = Translate(L, f)
M' = Translate(M, f)
N' = Translate(N, f)
L'' = Translate(L', z)
M'' = Translate(M', z)
N'' = Translate(N', z)
P = Translate(A'_2, z)
Q = Translate(B'_2, z)
R = Translate(C'_2, z)
A'_3 = Mirror(A, e)
B'_3 = Mirror(B, e)
C'_3 = Mirror(C, e)
poly1'_3, c'_3, a'_3, b'_3 = Polygon(A'_3, B'_3, C'_3)
S = Translate(A'_3, z)
T = Translate(B'_3, z)
U = Translate(C'_3, z)
poly3, u_1, s, t = Polygon(S, T, U)
step = 0
j = Segment(E, D)
bool1 = true
bool2 = false
V = Mirror(A'_1, d)
W = Mirror(B'_1, d)
Z = Mirror(C'_1, d)
poly4, z_1, v_1, w_1 = Polygon(V, W, Z)
A_1 = (8, 9)
k = Vector(A_1, D)
E' = Mirror(E, d)
l = Segment(E', D)
m = Vector(C'_3, D)
n = Segment(A, B)
p = Segment(B, C)
q = Segment(C, A)
C'_4 = Translate(C, w)
r = Vector(C, C'_4)
B_1 = Point(r)
a_1 = Vector(C, B_1)
A'_4 = Translate(A, a_1)
B'_4 = Translate(B, a_1)
C'_5 = Translate(C, a_1)
poly1'_4, c'_5, a'_4, b'_4 = Polygon(A'_4, B'_4, C'_5)
b_1 = Segment(B_1, C'_4)
A'_5 = Translate(A, r)
B'_5 = Translate(B, r)
n' = Segment(A'_5, B'_5)
B'_6 = Translate(B, r)
C'_6 = Translate(C, r)
p' = Segment(B'_6, C'_6)
C'_7 = Translate(C, r)
A'_6 = Translate(A, r)
q' = Segment(C'_7, A'_6)
C_1 = Mirror(A'_4, e)
D_1 = Mirror(B'_4, e)
E_1 = Mirror(C'_5, e)
poly5, e_1, c_1, d_1 = Polygon(C_1, D_1, E_1)
F_1 = (-6.5, 1.94)
G_1 = (-6.84, 9.18)
H_1 = (-0.36, 8.38)
f_1 = Segment(E, D)
I_1 = (5, 9)
J_1 = (5, 1)
K_1 = (6, 8)
L_1 = (6, 1.82)
M_1 = (6, 9)
N_1 = (6.02, 2.66)
g_1 = Line(M_1, N_1)
O_1 = (-6.2, 9.12)
h_1 = Segment(O, K)