Question
Which graphs represents the correct translation of right triangle DEF
on the coordinate plane with the following rule (x+3,y+1)
?(1 point)
Responses
A coordinate plane shows all four quadrants with both axes ranging from negative 8 to 8 in increments of 1. Two overlapping right triangles are graphed on the plane. Three points labeled as upper D, upper E, and upper F are plotted at coordinates left parenthesis negative 6 comma negative 5 right parenthesis, left parenthesis negative 6 comma negative 2 right parenthesis, and left parenthesis negative 3 comma negative 5 right parenthesis, respectively. The three points are joined by solid lines to form triangle upper D upper E upper F whose base is upper D upper F, hypotenuse is upper E upper F, and perpendicular side is upper D upper E. Three other points labeled as upper D prime, upper E prime, and upper F prime are plotted at coordinates left parenthesis negative 3 comma negative 5 right parenthesis, left parenthesis negative 3 comma negative 2 right parenthesis, and left parenthesis 0 comma negative 5 right parenthesis, respectively. These three points are joined by dotted lines to form triangle upper D prime upper E prime upper F prime whose base is upper D prime upper F prime, hypotenuse is upper E prime upper F prime, and perpendicular side is upper D prime upper E prime. The two triangles intersect at points upper F and upper D prime.
Image with alt text: A coordinate plane shows all four quadrants with both axes ranging from negative 8 to 8 in increments of 1. Two overlapping right triangles are graphed on the plane. Three points labeled as upper D, upper E, and upper F are plotted at coordinates left parenthesis negative 6 comma negative 5 right parenthesis, left parenthesis negative 6 comma negative 2 right parenthesis, and left parenthesis negative 3 comma negative 5 right parenthesis, respectively. The three points are joined by solid lines to form triangle upper D upper E upper F whose base is upper D upper F, hypotenuse is upper E upper F, and perpendicular side is upper D upper E. Three other points labeled as upper D prime, upper E prime, and upper F prime are plotted at coordinates left parenthesis negative 3 comma negative 5 right parenthesis, left parenthesis negative 3 comma negative 2 right parenthesis, and left parenthesis 0 comma negative 5 right parenthesis, respectively. These three points are joined by dotted lines to form triangle upper D prime upper E prime upper F prime whose base is upper D prime upper F prime, hypotenuse is upper E prime upper F prime, and perpendicular side is upper D prime upper E prime. The two triangles intersect at points upper F and upper D prime.
A coordinate plane shows all four quadrants with both axes ranging from negative 8 to 8 in increments of 1. Two right triangles are graphed on the plane. Three points labeled as upper D, upper E, and upper F are plotted at coordinates left parenthesis negative 6 comma negative 5 right parenthesis, left parenthesis negative 6 comma negative 2 right parenthesis, and left parenthesis negative 3 comma negative 5 right parenthesis, respectively. The three points are joined by solid lines to form triangle upper D upper E upper F whose base is upper D upper F, hypotenuse is upper E upper F, and perpendicular side is upper D upper E. Three other points labeled as upper D prime, upper E prime, and upper F prime are plotted at coordinates left parenthesis negative 3 comma negative 4 right parenthesis, left parenthesis negative 3 comma negative 1 right parenthesis, and left parenthesis 0 comma negative 4 right parenthesis, respectively. These three points are joined by dotted lines to form triangle upper D prime upper E prime upper F prime whose base is upper D prime upper F prime, hypotenuse is upper E prime upper F prime, and perpendicular side is upper D prime upper E prime.
Image with alt text: A coordinate plane shows all four quadrants with both axes ranging from negative 8 to 8 in increments of 1. Two right triangles are graphed on the plane. Three points labeled as upper D, upper E, and upper F are plotted at coordinates left parenthesis negative 6 comma negative 5 right parenthesis, left parenthesis negative 6 comma negative 2 right parenthesis, and left parenthesis negative 3 comma negative 5 right parenthesis, respectively. The three points are joined by solid lines to form triangle upper D upper E upper F whose base is upper D upper F, hypotenuse is upper E upper F, and perpendicular side is upper D upper E. Three other points labeled as upper D prime, upper E prime, and upper F prime are plotted at coordinates left parenthesis negative 3 comma negative 4 right parenthesis, left parenthesis negative 3 comma negative 1 right parenthesis, and left parenthesis 0 comma negative 4 right parenthesis, respectively. These three points are joined by dotted lines to form triangle upper D prime upper E prime upper F prime whose base is upper D prime upper F prime, hypotenuse is upper E prime upper F prime, and perpendicular side is upper D prime upper E prime.
A coordinate plane shows all four quadrants with both axes ranging from negative 8 to 8 in increments of 1. Two right triangles are graphed on the plane. Three points labeled as upper D, upper E, and upper F are plotted at coordinates left parenthesis negative 3 comma negative 4 right parenthesis, left parenthesis negative 3 comma negative 1 right parenthesis, and left parenthesis 0 comma negative 4 right parenthesis, respectively. The three points are joined by solid lines to form triangle upper D upper E upper F whose base is upper D upper F, hypotenuse is upper E upper F, and perpendicular side is upper D upper E. Three other points labeled as upper D prime, upper E prime, and upper F prime are plotted at coordinates left parenthesis negative 6 comma negative 5 right parenthesis, left parenthesis negative 6 comma negative 2 right parenthesis, and left parenthesis negative 3 comma negative 5 right parenthesis, respectively. These three points are joined by dotted lines to form triangle upper D prime upper E prime upper F prime whose base is upper D prime upper F prime, hypotenuse is upper E prime upper F prime, and perpendicular side is upper D prime upper E prime.
Image with alt text: A coordinate plane shows all four quadrants with both axes ranging from negative 8 to 8 in increments of 1. Two right triangles are graphed on the plane. Three points labeled as upper D, upper E, and upper F are plotted at coordinates left parenthesis negative 3 comma negative 4 right parenthesis, left parenthesis negative 3 comma negative 1 right parenthesis, and left parenthesis 0 comma negative 4 right parenthesis, respectively. The three points are joined by solid lines to form triangle upper D upper E upper F whose base is upper D upper F, hypotenuse is upper E upper F, and perpendicular side is upper D upper E. Three other points labeled as upper D prime, upper E prime, and upper F prime are plotted at coordinates left parenthesis negative 6 comma negative 5 right parenthesis, left parenthesis negative 6 comma negative 2 right parenthesis, and left parenthesis negative 3 comma negative 5 right parenthesis, respectively. These three points are joined by dotted lines to form triangle upper D prime upper E prime upper F prime whose base is upper D prime upper F prime, hypotenuse is upper E prime upper F prime, and perpendicular side is upper D prime upper E prime.
A coordinate plane shows all four quadrants with both axes ranging from negative 8 to 8 in increments of 1. Two right triangles are graphed on the plane. Three points labeled as upper D, upper E, and upper F are plotted at coordinates left parenthesis negative 6 comma negative 5 right parenthesis, left parenthesis negative 6 comma negative 2 right parenthesis, and left parenthesis negative 3 comma negative 5 right parenthesis, respectively. The three points are joined by solid lines to form triangle upper D upper E upper F whose base is upper D upper F, hypotenuse is upper E upper F, and perpendicular side is upper D upper E. Three other points labeled as upper D prime, upper E prime, and upper F prime are plotted at coordinates left parenthesis negative 5 comma negative 2 right parenthesis, left parenthesis negative 5 comma 1 right parenthesis, and left parenthesis negative 2 comma negative 2 right parenthesis, respectively. These three points are joined by dotted lines to form triangle upper D prime upper E prime upper F prime whose base is upper D prime upper F prime, hypotenuse is upper E prime upper F prime, and perpendicular side is upper D prime upper E prime.
on the coordinate plane with the following rule (x+3,y+1)
?(1 point)
Responses
A coordinate plane shows all four quadrants with both axes ranging from negative 8 to 8 in increments of 1. Two overlapping right triangles are graphed on the plane. Three points labeled as upper D, upper E, and upper F are plotted at coordinates left parenthesis negative 6 comma negative 5 right parenthesis, left parenthesis negative 6 comma negative 2 right parenthesis, and left parenthesis negative 3 comma negative 5 right parenthesis, respectively. The three points are joined by solid lines to form triangle upper D upper E upper F whose base is upper D upper F, hypotenuse is upper E upper F, and perpendicular side is upper D upper E. Three other points labeled as upper D prime, upper E prime, and upper F prime are plotted at coordinates left parenthesis negative 3 comma negative 5 right parenthesis, left parenthesis negative 3 comma negative 2 right parenthesis, and left parenthesis 0 comma negative 5 right parenthesis, respectively. These three points are joined by dotted lines to form triangle upper D prime upper E prime upper F prime whose base is upper D prime upper F prime, hypotenuse is upper E prime upper F prime, and perpendicular side is upper D prime upper E prime. The two triangles intersect at points upper F and upper D prime.
Image with alt text: A coordinate plane shows all four quadrants with both axes ranging from negative 8 to 8 in increments of 1. Two overlapping right triangles are graphed on the plane. Three points labeled as upper D, upper E, and upper F are plotted at coordinates left parenthesis negative 6 comma negative 5 right parenthesis, left parenthesis negative 6 comma negative 2 right parenthesis, and left parenthesis negative 3 comma negative 5 right parenthesis, respectively. The three points are joined by solid lines to form triangle upper D upper E upper F whose base is upper D upper F, hypotenuse is upper E upper F, and perpendicular side is upper D upper E. Three other points labeled as upper D prime, upper E prime, and upper F prime are plotted at coordinates left parenthesis negative 3 comma negative 5 right parenthesis, left parenthesis negative 3 comma negative 2 right parenthesis, and left parenthesis 0 comma negative 5 right parenthesis, respectively. These three points are joined by dotted lines to form triangle upper D prime upper E prime upper F prime whose base is upper D prime upper F prime, hypotenuse is upper E prime upper F prime, and perpendicular side is upper D prime upper E prime. The two triangles intersect at points upper F and upper D prime.
A coordinate plane shows all four quadrants with both axes ranging from negative 8 to 8 in increments of 1. Two right triangles are graphed on the plane. Three points labeled as upper D, upper E, and upper F are plotted at coordinates left parenthesis negative 6 comma negative 5 right parenthesis, left parenthesis negative 6 comma negative 2 right parenthesis, and left parenthesis negative 3 comma negative 5 right parenthesis, respectively. The three points are joined by solid lines to form triangle upper D upper E upper F whose base is upper D upper F, hypotenuse is upper E upper F, and perpendicular side is upper D upper E. Three other points labeled as upper D prime, upper E prime, and upper F prime are plotted at coordinates left parenthesis negative 3 comma negative 4 right parenthesis, left parenthesis negative 3 comma negative 1 right parenthesis, and left parenthesis 0 comma negative 4 right parenthesis, respectively. These three points are joined by dotted lines to form triangle upper D prime upper E prime upper F prime whose base is upper D prime upper F prime, hypotenuse is upper E prime upper F prime, and perpendicular side is upper D prime upper E prime.
Image with alt text: A coordinate plane shows all four quadrants with both axes ranging from negative 8 to 8 in increments of 1. Two right triangles are graphed on the plane. Three points labeled as upper D, upper E, and upper F are plotted at coordinates left parenthesis negative 6 comma negative 5 right parenthesis, left parenthesis negative 6 comma negative 2 right parenthesis, and left parenthesis negative 3 comma negative 5 right parenthesis, respectively. The three points are joined by solid lines to form triangle upper D upper E upper F whose base is upper D upper F, hypotenuse is upper E upper F, and perpendicular side is upper D upper E. Three other points labeled as upper D prime, upper E prime, and upper F prime are plotted at coordinates left parenthesis negative 3 comma negative 4 right parenthesis, left parenthesis negative 3 comma negative 1 right parenthesis, and left parenthesis 0 comma negative 4 right parenthesis, respectively. These three points are joined by dotted lines to form triangle upper D prime upper E prime upper F prime whose base is upper D prime upper F prime, hypotenuse is upper E prime upper F prime, and perpendicular side is upper D prime upper E prime.
A coordinate plane shows all four quadrants with both axes ranging from negative 8 to 8 in increments of 1. Two right triangles are graphed on the plane. Three points labeled as upper D, upper E, and upper F are plotted at coordinates left parenthesis negative 3 comma negative 4 right parenthesis, left parenthesis negative 3 comma negative 1 right parenthesis, and left parenthesis 0 comma negative 4 right parenthesis, respectively. The three points are joined by solid lines to form triangle upper D upper E upper F whose base is upper D upper F, hypotenuse is upper E upper F, and perpendicular side is upper D upper E. Three other points labeled as upper D prime, upper E prime, and upper F prime are plotted at coordinates left parenthesis negative 6 comma negative 5 right parenthesis, left parenthesis negative 6 comma negative 2 right parenthesis, and left parenthesis negative 3 comma negative 5 right parenthesis, respectively. These three points are joined by dotted lines to form triangle upper D prime upper E prime upper F prime whose base is upper D prime upper F prime, hypotenuse is upper E prime upper F prime, and perpendicular side is upper D prime upper E prime.
Image with alt text: A coordinate plane shows all four quadrants with both axes ranging from negative 8 to 8 in increments of 1. Two right triangles are graphed on the plane. Three points labeled as upper D, upper E, and upper F are plotted at coordinates left parenthesis negative 3 comma negative 4 right parenthesis, left parenthesis negative 3 comma negative 1 right parenthesis, and left parenthesis 0 comma negative 4 right parenthesis, respectively. The three points are joined by solid lines to form triangle upper D upper E upper F whose base is upper D upper F, hypotenuse is upper E upper F, and perpendicular side is upper D upper E. Three other points labeled as upper D prime, upper E prime, and upper F prime are plotted at coordinates left parenthesis negative 6 comma negative 5 right parenthesis, left parenthesis negative 6 comma negative 2 right parenthesis, and left parenthesis negative 3 comma negative 5 right parenthesis, respectively. These three points are joined by dotted lines to form triangle upper D prime upper E prime upper F prime whose base is upper D prime upper F prime, hypotenuse is upper E prime upper F prime, and perpendicular side is upper D prime upper E prime.
A coordinate plane shows all four quadrants with both axes ranging from negative 8 to 8 in increments of 1. Two right triangles are graphed on the plane. Three points labeled as upper D, upper E, and upper F are plotted at coordinates left parenthesis negative 6 comma negative 5 right parenthesis, left parenthesis negative 6 comma negative 2 right parenthesis, and left parenthesis negative 3 comma negative 5 right parenthesis, respectively. The three points are joined by solid lines to form triangle upper D upper E upper F whose base is upper D upper F, hypotenuse is upper E upper F, and perpendicular side is upper D upper E. Three other points labeled as upper D prime, upper E prime, and upper F prime are plotted at coordinates left parenthesis negative 5 comma negative 2 right parenthesis, left parenthesis negative 5 comma 1 right parenthesis, and left parenthesis negative 2 comma negative 2 right parenthesis, respectively. These three points are joined by dotted lines to form triangle upper D prime upper E prime upper F prime whose base is upper D prime upper F prime, hypotenuse is upper E prime upper F prime, and perpendicular side is upper D prime upper E prime.
Answers
GPT-4o mini
To determine the correct translation of right triangle DEF according to the rule \((x + 3, y + 1)\), we need to apply the translation to each vertex of triangle DEF.
Given the original coordinates of points D, E, and F:
- D: \((-6, -5)\)
- E: \((-6, -2)\)
- F: \((-3, -5)\)
Now, apply the translation to each point:
1. **Translation of D**:
\[
(-6 + 3, -5 + 1) = (-3, -4)
\]
2. **Translation of E**:
\[
(-6 + 3, -2 + 1) = (-3, -1)
\]
3. **Translation of F**:
\[
(-3 + 3, -5 + 1) = (0, -4)
\]
Now we have the translated coordinates:
- \(D' = (-3, -4)\)
- \(E' = (-3, -1)\)
- \(F' = (0, -4)\)
Now, let's find the graph that represents the new triangle \(D'E'F'\) with these coordinates.
Looking at the options:
- The triangle represented by the coordinates \((-3, -4)\), \((-3, -1)\), and \((0, -4)\) must match one of the given options.
Thus, the triangle with vertices at:
- \(D' = (-3, -4)\)
- \(E' = (-3, -1)\)
- \(F' = (0, -4)\)
is found in the options provided. This should allow you to correctly identify the graph that corresponds to the translated triangle.
Given the original coordinates of points D, E, and F:
- D: \((-6, -5)\)
- E: \((-6, -2)\)
- F: \((-3, -5)\)
Now, apply the translation to each point:
1. **Translation of D**:
\[
(-6 + 3, -5 + 1) = (-3, -4)
\]
2. **Translation of E**:
\[
(-6 + 3, -2 + 1) = (-3, -1)
\]
3. **Translation of F**:
\[
(-3 + 3, -5 + 1) = (0, -4)
\]
Now we have the translated coordinates:
- \(D' = (-3, -4)\)
- \(E' = (-3, -1)\)
- \(F' = (0, -4)\)
Now, let's find the graph that represents the new triangle \(D'E'F'\) with these coordinates.
Looking at the options:
- The triangle represented by the coordinates \((-3, -4)\), \((-3, -1)\), and \((0, -4)\) must match one of the given options.
Thus, the triangle with vertices at:
- \(D' = (-3, -4)\)
- \(E' = (-3, -1)\)
- \(F' = (0, -4)\)
is found in the options provided. This should allow you to correctly identify the graph that corresponds to the translated triangle.