The vertices of ΔABC are A (1, 5), B (3, 9), and C (5, 3). The vertices of ΔDEF are D (−3, 3), E (−2, 5), and F (−1, 2). Which conclusion is true about the triangles?
The ratio of their corresponding sides is 1:3.
They are congruent by the definition of congruence in terms of rigid motions.
The ratio of their corresponding angles is 1:3.
They are similar by the definition of similarity in terms of a dilation.
AB = √20
DE = √5
So, they are not congruent.
If they are similar, the ratio is not 1:3
so I'd guess rigid motions. Check the other sides to be sure.
choice C is just stupid. Similar triangles have the same angles.
Answer choice "D" is correct
They are similar by the definition of similarity in terms of a dilation.
To determine the conclusion that is true about the triangles, we need to explore the given information.
Let's begin by calculating the lengths of the sides of ΔABC and ΔDEF.
The distance between points A(1, 5) and B(3, 9) can be found using the distance formula:
d₁ = √((x₂ - x₁)² + (y₂ - y₁)²)
d₁ = √((3 - 1)² + (9 - 5)²)
d₁ = √(2² + 4²)
d₁ = √(4 + 16)
d₁ = √20
d₁ ≈ 4.47
Similarly, the distances between points B and C, and between points C and A can be calculated:
d₂ (BC) ≈ 4.47
d₃ (CA) ≈ 4.47
Now, let's calculate the lengths of the sides of ΔDEF.
The distance between points D(-3, 3) and E(-2, 5) can be found using the distance formula:
d₄ = √((-2 - (-3))² + (5 - 3)²)
d₄ = √(1² + 2²)
d₄ = √(1 + 4)
d₄ = √5
d₄ ≈ 2.24
Similarly, the distances between points E and F, and between points F and D can be calculated:
d₅ (EF) ≈ 2.24
d₆ (FD) ≈ 2.24
Now, let's compare the corresponding sides:
d₁ (AB) ≈ 4.47
d₄ (DE) ≈ 2.24
d₂ (BC) ≈ 4.47
d₅ (EF) ≈ 2.24
d₃ (CA) ≈ 4.47
d₆ (FD) ≈ 2.24
By comparing the corresponding sides of the two triangles, we can see that their lengths are not equal and do not have a constant ratio. Therefore, the conclusion that the ratio of their corresponding sides is 1:3 is not true.
Let's move on to check the congruence of the triangles by the definition of congruence through rigid motions.
To determine if the triangles are congruent, we need to check if one triangle can be transformed into the other using translations, rotations, and/or reflections.
By visually inspecting the positions and orientations of the triangles, we can see that they do not appear to be congruent. Therefore, the conclusion that they are congruent by the definition of congruence in terms of rigid motions is not true.
Now, let's check if the triangles are similar by the definition of similarity in terms of a dilation.
For two triangles to be similar, the corresponding angles need to have the same measures, and the corresponding sides need to have a constant ratio.
To determine if the angles are proportional, we can compare the measures of the corresponding angles:
∠A ≈ ∠D ≈ 63.43°
∠B ≈ ∠E ≈ 45°
∠C ≈ ∠F ≈ 71.57°
The measures of the angles are not proportional, so the conclusion that the ratio of their corresponding angles is 1:3 is not true.
Based on these calculations and comparisons, it can be concluded that none of the given conclusions about the triangles are true.
To determine which conclusion is true about the triangles, we need to analyze their corresponding sides and corresponding angles.
Let's start by finding the lengths of the sides of triangle ABC and triangle DEF:
Triangle ABC:
Side AB = √[(x₂ - x₁)² + (y₂ - y₁)²] = √[(3 - 1)² + (9 - 5)²] = √[2² + 4²] = √20
Side BC = √[(x₂ - x₁)² + (y₂ - y₁)²] = √[(5 - 3)² + (3 - 9)²] = √[2² + (-6)²] = √40
Side AC = √[(x₂ - x₁)² + (y₂ - y₁)²] = √[(5 - 1)² + (3 - 5)²] = √[4² + (-2)²] = √20
Triangle DEF:
Side DE = √[(x₂ - x₁)² + (y₂ - y₁)²] = √[(-2 - (-3))² + (5 - 3)²] = √[1² + 2²] = √5
Side EF = √[(x₂ - x₁)² + (y₂ - y₁)²] = √[(-1 - (-2))² + (2 - 5)²] = √[1² + (-3)²] = √10
Side DF = √[(x₂ - x₁)² + (y₂ - y₁)²] = √[(-1 - (-3))² + (2 - 3)²] = √[2² + 1²] = √5
Now, let's compare the ratios of the corresponding sides:
Ratio of sides AB and DE: (√20) / (√5) = (√(4 * 5)) / (√5) = √4 = 2
Ratio of sides BC and EF: (√40) / (√10) = (√(4 * 10)) / (√10) = √4 = 2
Ratio of sides AC and DF: (√20) / (√5) = (√(4 * 5)) / (√5) = √4 = 2
The ratios of the corresponding sides are all equal to 2:1, which means the first option, "The ratio of their corresponding sides is 1:3," is false.
Now, let's analyze the corresponding angles. To calculate the angles, we can use the slope formula:
Triangle ABC:
Slope of AB = (y₂ - y₁) / (x₂ - x₁) = (9 - 5) / (3 - 1) = 4 / 2 = 2
Slope of BC = (y₂ - y₁) / (x₂ - x₁) = (3 - 9) / (5 - 3) = -6 / 2 = -3
Slope of AC = (y₂ - y₁) / (x₂ - x₁) = (3 - 5) / (5 - 1) = -2 / 4 = -1/2
Triangle DEF:
Slope of DE = (y₂ - y₁) / (x₂ - x₁) = (5 - 3) / (-2 - (-3)) = 2 / 1 = 2
Slope of EF = (y₂ - y₁) / (x₂ - x₁) = (2 - 5) / (-1 - (-2)) = -3 / 1 = -3
Slope of DF = (y₂ - y₁) / (x₂ - x₁) = (2 - 3) / (-1 - (-3)) = 1 / 2 = 1/2
From the slopes, we can see that no two corresponding lines have the same slope, which means the triangles are not congruent by the definition of congruence in terms of rigid motions. Thus, the second option, "They are congruent by the definition of congruence in terms of rigid motions," is false.
Finally, let's compare the ratios of the corresponding angles:
Ratio of angles A and D: None of the lines have the same slope, so the angles cannot be compared directly.
Ratio of angles B and E: None of the lines have the same slope, so the angles cannot be compared directly.
Ratio of angles C and F: None of the lines have the same slope, so the angles cannot be compared directly.
Since we cannot determine the ratios of corresponding angles, the third option, "The ratio of their corresponding angles is 1:3," is also false.
This leaves us with the fourth option, "They are similar by the definition of similarity in terms of a dilation." Since the ratios of the corresponding sides are all equal, we can conclude that the triangles are similar by the definition of similarity in terms of a dilation.