The quilt shown has a border made of righta. Graph one triangle with vertices at D(-4, 0), E(0, 0), and F(0, -8). To locate
another triangle, follow these steps:
Reflect ADEF across the y-axis to form AD'E'F'.
Translate AD'E'F' 8 units up to form AD"E"F".
Is ADEF the same size and shape as AD"E"F"? Show your
.
.
x
work.

Answer:

To determine if triangles ADEF and AD"E"F" are the same size and shape, we need to compare their corresponding sides and angles. Let's go through the steps and analyze each transformation.

1. Reflecting ADEF across the y-axis:

When we reflect a point across the y-axis, the x-coordinate changes sign while the y-coordinate remains the same. Applying this reflection to each vertex of triangle ADEF, we get D'(-4, 0), E'(0, 0), and F'(0, -8).

2. Translating AD'E'F' 8 units up:

To translate a point, we add or subtract a constant value to its coordinates. Shifting each vertex of triangle AD'E'F' 8 units up gives us D"( -4, 8), E"(0, 8), and F"(0, 0).

Now let's compare the corresponding sides and angles of triangles ADEF and AD"E"F".

Corresponding Sides:

Side DE in triangle ADEF corresponds to side D'E' in triangle AD'E'F'. Since both sides lie on the x-axis, they have the same length.

Side EF in triangle ADEF corresponds to side E'F' in triangle AD'E'F'. Again, both sides lie on the y-axis, so they have the same length.

Side FD in triangle ADEF corresponds to side F'D' in triangle AD'E'F'. These sides are vertical lines with equal lengths.

Corresponding Angles:

Angle D in triangle ADEF corresponds to angle D' in triangle AD'E'F'. Both angles are right angles (90 degrees).

Angle E in triangle ADEF corresponds to angle E' in triangle AD'E'F'. These angles are also right angles (90 degrees).

Angle F in triangle ADEF corresponds to angle F' in triangle AD'E'F'. Once again, these angles are right angles (90 degrees).

From the above analysis, we can conclude that triangles ADEF and AD"E"F" are indeed the same size and shape. They have corresponding sides of equal length and corresponding angles of equal measure.

In conclusion, triangles ADEF and AD"E"F" are congruent.

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