the vertices of triangle DEF are D(5,12)and E(2,7) and F(8,4)
Triangle DEF undergoes an enlargement with the centre ,O, and scale factors k.Its image is D`E`F` where D(5,12)arrow D`(7.5,18)
a)How do i detremine the value of k
To determine the value of k, we can use the distance formula. The distance between two points (x1, y1) and (x2, y2) is given by:
d = √[(x2 - x1)^2 + (y2 - y1)^2]
In this case, we can use the distance between the original point D(5, 12) and its image point D`(7.5, 18) to find the value of k.
Using the distance formula:
d = √[(x2 - x1)^2 + (y2 - y1)^2]
d = √[(7.5 - 5)^2 + (18 - 12)^2]
d = √[(2.5)^2 + (6)^2]
d = √[6.25 + 36]
d = √42.25
d ≈ 6.50
The scale factor, k, is the ratio of the distance between the original point and its image point to the distance between the center of enlargement O and the original point.
Using the distance formula:
k = d / √[(x2 - x1)^2 + (y2 - y1)^2]
k = 6.50 / √[(8 - 5)^2 + (4 - 12)^2]
k = 6.50 / √[(3)^2 + (-8)^2]
k = 6.50 / √[9 + 64]
k = 6.50 / √73
k ≈ 0.758
Therefore, the value of k is approximately 0.758.
To determine the value of k, you can use the concept that the scale factor is the ratio of the corresponding side lengths of the pre-image triangle (DEF) and the image triangle (D`E`F`). In this case, since we have the coordinates of D and D`, we can use their distances to find the value of k.
First, let's calculate the distance between D and D` using the distance formula:
Distance between two points (x₁, y₁) and (x₂, y₂) can be calculated as:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
In this case, D(5,12) and D`(7.5,18), so the distance between them is:
d = √((7.5 - 5)² + (18 - 12)²)
= √((2.5)² + (6)²)
= √(6.25 + 36)
= √42.25
≈ 6.5
Next, we can calculate the distance between D and E, which represents the corresponding side lengths in the pre-image and image triangles:
Distance between D and E can be calculated as:
d₁ = √((x₂ - x₁)² + (y₂ - y₁)²)
= √((2 - 5)² + (7 - 12)²)
= √((-3)² + (-5)²)
= √(9 + 25)
= √34
≈ 5.83
Finally, divide the distance between D and D` by the distance between D and E to find the scale factor (k):
k = d / d₁
= 6.5 / 5.83
≈ 1.12
So, the approximate value of k is 1.12.
It looks to me like the new triangle is 1.5 times larger. All primed coordinates are 1.5 times the original. The center of the expansion is the origin (O).
The answer is k=1.5