Triangle ABC has vertices of A(0,0), B(–4,0)and C(–2,4). The coordinates of each vertex in triangle AEF are multiplied by 3 Is triangle ABC~AEF? Explain.
To determine if triangles ABC and AEF are similar, we need to compare their corresponding sides and angles.
Let's find the coordinates of the vertices of triangle AEF after multiplying each coordinate in triangle ABC by 3.
Vertex A in triangle AEF: (3(0), 3(0)) = (0, 0)
Vertex B in triangle AEF: (3(-4), 3(0)) = (-12, 0)
Vertex C in triangle AEF: (3(-2), 3(4)) = (-6, 12)
Now, let's compare the lengths of the corresponding sides:
Side AB in triangle ABC: Distance between A(0, 0) and B(-4, 0) is 4 units.
Side AE in triangle AEF: Distance between A(0, 0) and E(-12, 0) is 12 units.
Side BC in triangle ABC: Distance between B(-4, 0) and C(-2, 4) is 4 units.
Side EF in triangle AEF: Distance between E(-12, 0) and F(-6, 12) is also 12 units.
Side AC in triangle ABC: Distance between A(0, 0) and C(-2, 4) is approximately 4.47 units.
Side AF in triangle AEF: Distance between A(0, 0) and F(-6, 12) is approximately 12.73 units.
Looking at the corresponding sides, we can see that the lengths of the sides in triangle AEF are all multiplied by 3 compared to triangle ABC. This means that the lengths of the corresponding sides are proportional, which is one of the criteria for similarity.
Therefore, we can conclude that triangle ABC and triangle AEF are similar because their corresponding side lengths are proportional.
To determine if triangle ABC is similar to triangle AEF, we need to compare the ratios of corresponding sides.
Step 1: Find the distances between the vertices of triangle ABC.
- Distance between A and B: √[(x₂ - x₁)² + (y₂ - y₁)²]
= √[(-4 - 0)² + (0 - 0)²] = √[16] = 4
- Distance between B and C: √[(x₂ - x₁)² + (y₂ - y₁)²]
= √[(-2 - (-4))² + (4 - 0)²] = √[4 + 16] = √20 = 2√5
- Distance between C and A: √[(x₂ - x₁)² + (y₂ - y₁)²]
= √[(0 - (-2))² + (0 - 4)²] = √[4 + 16] = √20 = 2√5
Step 2: Find the distances between the corresponding vertices of triangle AEF.
- Distance between A' and E: √[(x₂ - x₁)² + (y₂ - y₁)²]
= √[(0 - 0)² + (0 - 0)²] = √[0] = 0
- Distance between E and F: √[(x₂ - x₁)² + (y₂ - y₁)²]
= √[(-12 - 0)² + (0 - 0)²] = √[144] = 12
- Distance between F and A': √[(x₂ - x₁)² + (y₂ - y₁)²]
= √[(0 - (-12))² + (0 - 0)²] = √[144] = 12
Step 3: Compare the ratios of corresponding side lengths.
The ratio of the side lengths of triangle AEF to triangle ABC can be calculated as follows:
- A' to E ratio: 0 : 4 = 0
- E to F ratio: 12 : 2√5 = 6 : √5
- F to A' ratio: 12 : 4 = 3
The ratio of the side lengths of triangle AEF does not match the ratio of the side lengths of triangle ABC, indicating that the triangles are not similar.
Therefore, triangle ABC is not similar to triangle AEF.
yes
The parallelism of the corresponding line is maintained, so the angles stay the same
e.g. B(-4,0) ----> B1(-12,0)
C(-2,4) -------> C1(-6,12)
slope AB = (4-0)/(-2+4) = 4/2 = 2
slope A1B1=(12-0)/(-6+12) = 12/6 =2