Question

18
select the correct answer from each drop - down menu.
△abc has vertices at a(12,8), b(4,8), and c(4,14).
△xyz has vertices at x(6,6), y(4,12), and z(10,14).
△mno has vertices at m(4,16), n(4,8), and o(-2,8).
△jkl has vertices at j(14,-2), k(12,2), and l(20,4).
are congruent a is a single rigid transformation that maps the two congruent triangles.

Explanation:

Step1: Analyze Triangle ABC

First, find the side lengths of \( \triangle ABC \). The coordinates are \( A(12, 8) \), \( B(4, 8) \), \( C(4, 14) \).
- Length of \( AB \): Since \( A \) and \( B \) have the same \( y \)-coordinate, \( AB = |12 - 4| = 8 \).
- Length of \( BC \): Since \( B \) and \( C \) have the same \( x \)-coordinate, \( BC = |14 - 8| = 6 \).
- Length of \( AC \): Using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), \( AC = \sqrt{(4 - 12)^2 + (14 - 8)^2} = \sqrt{(-8)^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \). So \( \triangle ABC \) is a right triangle with legs 6 and 8, hypotenuse 10.

Step2: Analyze Triangle MNO

Coordinates of \( M(4, 16) \), \( N(4, 8) \), \( O(-2, 8) \).
- Length of \( MN \): Same \( x \)-coordinate, \( MN = |16 - 8| = 8 \).
- Length of \( NO \): Same \( y \)-coordinate, \( NO = |4 - (-2)| = 6 \).
- Length of \( MO \): Distance formula: \( MO = \sqrt{(-2 - 4)^2 + (8 - 16)^2} = \sqrt{(-6)^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \). So \( \triangle MNO \) is a right triangle with legs 6 and 8, hypotenuse 10. So \( \triangle ABC \) and \( \triangle MNO \) are congruent (SSS congruence, since all sides are equal).

Step3: Analyze other triangles (for elimination)

- \( \triangle XYZ \): Coordinates \( X(6,6) \), \( Y(4,12) \), \( Z(10,14) \). Let's find side lengths. \( XY = \sqrt{(4 - 6)^2 + (12 - 6)^2} = \sqrt{4 + 36} = \sqrt{40} \), \( YZ = \sqrt{(10 - 4)^2 + (14 - 12)^2} = \sqrt{36 + 4} = \sqrt{40} \), \( XZ = \sqrt{(10 - 6)^2 + (14 - 6)^2} = \sqrt{16 + 64} = \sqrt{80} \). Not matching \( ABC \)'s sides.
- \( \triangle JKL \): Coordinates \( J(14,-2) \), \( K(12,2) \), \( L(20,4) \). Side lengths: \( JK = \sqrt{(12 - 14)^2 + (2 - (-2))^2} = \sqrt{4 + 16} = \sqrt{20} \), \( KL = \sqrt{(20 - 12)^2 + (4 - 2)^2} = \sqrt{64 + 4} = \sqrt{68} \), \( JL = \sqrt{(20 - 14)^2 + (4 - (-2))^2} = \sqrt{36 + 36} = \sqrt{72} \). Not matching \( ABC \)'s sides.

So the congruent triangles are \( \triangle ABC \) and \( \triangle MNO \). Now, for the rigid transformation: Let's see the positions. \( \triangle ABC \) has \( B(4,8) \), \( C(4,14) \), \( A(12,8) \). \( \triangle MNO \) has \( N(4,8) \), \( O(-2,8) \), \( M(4,16) \). Notice that \( \triangle MNO \) is a reflection (or rotation) of \( \triangle ABC \)? Wait, actually, if we reflect \( \triangle ABC \) over the y-axis? Wait, no. Wait, \( \triangle ABC \): points \( A(12,8) \), \( B(4,8) \), \( C(4,14) \). \( \triangle MNO \): \( M(4,16) \), \( N(4,8) \), \( O(-2,8) \). Let's check the transformation. The base \( AB \) is from \( x=4 \) to \( x=12 \) (right), \( NO \) is from \( x=-2 \) to \( x=4 \) (left), but the length is same. The height \( BC \) is from \( y=8 \) to \( y=14 \) (up), \( MN \) is from \( y=8 \) to \( y=16 \) (up). Wait, actually, a reflection over the line \( x = 4 \)? No, maybe a rotation. Alternatively, since both are right triangles with right angle at \( B \) (for \( ABC \), right angle at \( B \) because \( AB \) horizontal, \( BC \) vertical) and right angle at \( N \) (for \( MNO \), \( NO \) horizontal, \( MN \) vertical). So the right angle is at \( B(4,8) \) and \( N(4,8) \). So \( B \) and \( N \) are the same point? Wait, \( B \) is \( (4,8) \), \( N \) is \( (4,8) \). Oh! So \( B \) and \( N \) coincide. Then \( BC \) is from \( (4,8) \) to \( (4,14) \) (up 6), \( MN \) is from \( (4,8) \) to \( (4,16) \) (up 8)? Wait no, earlier calculation: \( MN \) length is 8, \( BC \) length is 6. Wait, no, earlier mistake? Wait no, \( BC \) is from \( (4,8) \) to \( (4,14) \), so length 6 (14-8=6). \( MN \) is from \( (4,8) \) to \( (4,16) \), length 8 (16-8=8). Wait, no, earlier calculation for \( MN \): \( M(4,16) \), \( N(4,8) \), so distance is 8, correct. \( BC \): \( B(4,8) \), \( C(4,14) \), distance 6, correct. So right angle at \( B \) (for \( ABC \)): \( AB \) is horizontal (from \( (12,8) \) to \( (4,8) \), length 8), \( BC \) vertical (length 6). Right angle at \( N \) (for \( MNO \)): \( NO \) is horizontal (from \( (-2,8) \) to \( (4,8) \), length 6), \( MN \) vertical (from \( (4,8) \) to \( (4,16) \), length 8). So the right angle sides are swapped (8 and 6). So a rotation of 90 degrees? Wait, no, the sides are length 8 and 6, so swapping the legs would be a rotation of 90 degrees. Alternatively, a reflection and a rotation. But the key is that the congruent triangles are \( ABC \) and \( MNO \), and a possible rigid transformation is a rotation (or reflection). But the first part is identifying the congruent triangles: \( \triangle ABC \) and \( \triangle MNO \).

Answer:

First drop - down: Triangle ABC and triangle MNO
Second drop - down: (Possible rigid transformations: rotation, reflection, translation; since they share the point \( (4,8) \), a rotation about \( (4,8) \) by 90 degrees, or reflection. But the main part here is the first drop - down. So the first answer is "Triangle ABC and triangle MNO".)