Triangle XYZ is translated down 4 units and to the left 8 units. The length of side XY is 10 units. What is the length of side X′Y′?(1 point)
To find the length of side X'Y', we need to understand the translation of triangle XYZ.
The translation states that the triangle is moved down 4 units and to the left 8 units. This means that every point of the triangle is shifted down by 4 units and to the left by 8 units.
Let's label the vertices of triangle XYZ:
X = (x, y), Y = (x, y + 10), and Z = (x + ?, y + ?).
Note that we know the length of side XY, which is 10 units. This implies that the y-coordinate of vertex Y is y + 10.
Since the triangle is translated down 4 units, the y-coordinate of each vertex is decreased by 4:
Y' = (x, y + 10 - 4) = (x, y + 6).
Similarly, since the triangle is translated to the left 8 units, the x-coordinate of each vertex is decreased by 8:
X' = (x - 8, y) and Z' = (x - 8 + ?, y + ?).
Now, we need to find the length of side X'Y'.
Using the coordinates of X' and Y', we can calculate the length of X'Y' using the distance formula:
distance = √((x₂ - x₁)² + (y₂ - y₁)²),
where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points.
Substituting the coordinates of X' and Y' into the distance formula, we get:
length of X'Y' = √(((x - 8) - x)² + ((y + 6) - y)²).
Simplifying this expression:
length of X'Y' = √((-8)² + 6²) = √(64 + 36) = √100 = 10 units.
Therefore, the length of side X'Y' is 10 units.
To find the length of side X'Y', we need to understand the concept of translation and how it affects the coordinates of a triangle.
In this case, the triangle XYZ is translated down 4 units and to the left 8 units. This means that each point of the original triangle will be shifted by 4 units downward and 8 units to the left.
Given that XY has a length of 10 units, we can determine the new coordinates of the triangle after translation.
Let's assume the coordinates of point X are (X1, X2), the coordinates of point Y are (Y1, Y2), and the coordinates of point Z are (Z1, Z2).
After the translation, the new coordinates of the triangle will be:
X' = (X1 - 8, X2 - 4)
Y' = (Y1 - 8, Y2 - 4)
Z' = (Z1 - 8, Z2 - 4)
Since we are interested in the length of side X'Y', we can compute it using the distance formula:
Distance = sqrt((X'2 - X'1)^2 + (Y'2 - Y'1)^2)
Substituting the coordinates we found earlier, we get:
Distance = sqrt((Y1 - 8 - X1 + 8)^2 + (Y2 - 4 - X2 + 4)^2)
Simplifying this expression:
Distance = sqrt((Y1 - X1)^2 + (Y2 - X2)^2)
Therefore, the length of side X'Y' is equal to the distance between points X and Y, which is 10 units.
WRONG
Translations do not change the lengths of corresponding sides, so ....