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. 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.
B. 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.
C. 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.
D. 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. 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.
B. 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.
C. 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.
D. 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 which graph correctly represents the translation of right triangle DEF according to the rule (x+3, y+1), we need to apply the translation to the coordinates of each point in triangle DEF, which we have at the coordinates:
- D = (-6, -5)
- E = (-6, -2)
- F = (-3, -5)
Using the translation rule, we will add 3 to the x-coordinates and 1 to the y-coordinates of each point:
1. **Translate point D**:
- D' = (-6 + 3, -5 + 1) = (-3, -4)
2. **Translate point E**:
- E' = (-6 + 3, -2 + 1) = (-3, -1)
3. **Translate point F**:
- F' = (-3 + 3, -5 + 1) = (0, -4)
So the translated coordinates are:
- D' = (-3, -4)
- E' = (-3, -1)
- F' = (0, -4)
Now we'll compare these results to the options provided:
- **A:** D' = (-3, -5), E' = (-3, -2), F' = (0, -5) (Incorrect)
- **B:** D' = (-3, -4), E' = (-3, -1), F' = (0, -4) (Correct)
- **C:** D' = (-3, -4), E' = (-3, -1), F' = (0, -4) (also correct, but not the same triangle as B because original triangle DEF is at different coordinates; B is valid one)
- **D:** D' = (-5, -2), E' = (-5, 1), F' = (-2, -2) (Incorrect)
From this, graph **B** accurately represents the translation of triangle DEF according to the specified rule.
- D = (-6, -5)
- E = (-6, -2)
- F = (-3, -5)
Using the translation rule, we will add 3 to the x-coordinates and 1 to the y-coordinates of each point:
1. **Translate point D**:
- D' = (-6 + 3, -5 + 1) = (-3, -4)
2. **Translate point E**:
- E' = (-6 + 3, -2 + 1) = (-3, -1)
3. **Translate point F**:
- F' = (-3 + 3, -5 + 1) = (0, -4)
So the translated coordinates are:
- D' = (-3, -4)
- E' = (-3, -1)
- F' = (0, -4)
Now we'll compare these results to the options provided:
- **A:** D' = (-3, -5), E' = (-3, -2), F' = (0, -5) (Incorrect)
- **B:** D' = (-3, -4), E' = (-3, -1), F' = (0, -4) (Correct)
- **C:** D' = (-3, -4), E' = (-3, -1), F' = (0, -4) (also correct, but not the same triangle as B because original triangle DEF is at different coordinates; B is valid one)
- **D:** D' = (-5, -2), E' = (-5, 1), F' = (-2, -2) (Incorrect)
From this, graph **B** accurately represents the translation of triangle DEF according to the specified rule.