QuestionTriangle ABC with vertices A(4,2), B(2,7), C(6,5) is translated. Which transformations map it to A''B''C''? Options: A, B, or C?

Studdy Solution

STEP 1

Assumptions1. The vertices of triangle $ABC$ are $A(4,), B(,7)$ , and $C(6,5)$ . . The triangle is translated6 units left and5 units down.
3. We need to find the sequence of transformations that maps $\triangle ABC$ onto $\triangle A''B''C''$ .

STEP 2

First, we need to apply the translation to the coordinates of the vertices of the triangle. The translation rule for moving a point $p(x, y)$ left by $a$ units and down by $b$ units is $p'(x-a, y-b)$ .

STEP 3

Apply the translation rule to the coordinates of the vertices of the triangle.
$A'(-6,2-5)$ $B'(2-6,7-5)$ $C'(6-6,5-5)$

STEP 4

Calculate the new coordinates for the vertices of the triangle after the translation.
$A'(-2, -3)$ $B'(-4,2)$ $C'(0,0)$

STEP 5

Now we need to apply the transformations given in the options to the translated triangle and see which one maps $\triangle A'B'C'$ onto $\triangle A''B''C''$ .

STEP 6

Option A suggests a $90^{\circ}$ counterclockwise rotation about the origin followed by a reflection over the line $y=2$ . The rule for a $90^{\circ}$ counterclockwise rotation about the origin is $p'(y, -x)$ and the rule for a reflection over the line $y=k$ is $p'(x,2k-y)$ .

STEP 7

Apply the rotation rule to the coordinates of the vertices of the translated triangle.
$A''(A'_y, -A'_x)$ $B''(B'_y, -B'_x)$ $C''(C'_y, -C'_x)$

STEP 8

Calculate the new coordinates for the vertices of the triangle after the rotation.
$A''(-3,2)$ $B''(2,4)$ $C''(0,0)$

STEP 9

Apply the reflection rule to the coordinates of the vertices of the rotated triangle.
$A'''(A''x,22 - A''y)$ $B'''(B''x,22 - B''y)$ $C'''(C''x,2*2 - C''y)$

STEP 10

Calculate the new coordinates for the vertices of the triangle after the reflection.
$A'''(-3,7)$ $B'''(2,0)$ $C'''(0,4)$

STEP 11

Repeat steps6 to10 for options B and C.

STEP 12

Compare the final coordinates for the vertices of the triangle after each sequence of transformations with the coordinates of the vertices of $\triangle A''B''C''$ to determine which sequence of transformations maps $\triangle ABC$ onto $\triangle A''B''C''$ .
The solution will be the sequence of transformations that results in the same coordinates for the vertices of the triangle as $\triangle A''B''C''$ .

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