Question

Triangle upper X upper Y upper Z has vertices located as follows: upper X at left parenthesis negative 3 comma negative 3 right parenthesis, upper Y at left parenthesis negative 3 comma negative 6 right parenthesis, and upper Z at left parenthesis negative 6 comma negative 3 right parenthesis. Triangle upper X prime upper Y prime upper Z prime has vertices located as follows: upper X prime at left parenthesis negative 1 comma negative 1 right parenthesis, upper Y prime at left parenthesis negative 1 comma negative 2 right parenthesis, and upper Z prime at left parenthesis negative 2 comma negative 1 right parenthesis.

Which statement proves that the dilation of ΔXYZ
Δ
𝑋
𝑌
𝑍
does not pass through the center of dilation if the center of dilation is the origin?

X′(−3⋅13, −3⋅13)=X′(−1, −1) 𝑋 ′ ( − 3 ⋅ 1 3 , − 3 ⋅ 1 3 ) = 𝑋 ′ ( − 1 , − 1 ) , Y′(−3⋅13, −6⋅13)=Y′(−1, −2) 𝑌 ′ ( − 3 ⋅ 1 3 , − 6 ⋅ 1 3 ) = 𝑌 ′ ( − 1 , − 2 ) , Z′(−6⋅13, −3⋅13)=Z′(−2, −1) 𝑍 ′ ( − 6 ⋅ 1 3 , − 3 ⋅ 1 3 ) = 𝑍 ′ ( − 2 , − 1 ) .
XY¯¯¯¯¯¯¯¯
𝑋
𝑌
¯
has a length of 3 and X′Y′¯¯¯¯¯¯¯¯¯¯¯¯
𝑋
′
𝑌
′
¯
has a length of 1, which proves that ΔX′Y′Z′
Δ
𝑋
′
𝑌
′
𝑍
′
is a dilation of ΔXYZ
Δ
𝑋
𝑌
𝑍
by a scale factor of 13
1
3
.
Modifying above upper X upper Y with bar has a length of 3 and Modifying above upper X prime upper Y prime with bar has a length of 1, which proves that triangle upper X prime upper Y prime upper Z prime is a dilation of triangle upper X upper Y upper Z by a scale factor of Start Fraction 1 over 3 End Fraction .

XY¯¯¯¯¯¯¯¯
𝑋
𝑌
¯
has a length of 3 and X′Y′¯¯¯¯¯¯¯¯¯¯¯¯
𝑋
′
𝑌
′
¯
has a length of 1, which proves that ΔXYZ
Δ
𝑋
𝑌
𝑍
is a dilation of ΔX′Y′Z′
Δ
𝑋
′
𝑌
′
𝑍
′
by a scale factor of 3.
Modifying above upper X upper Y with bar has a length of 3 and Modifying above upper X prime upper Y prime with bar has a length of 1, which proves that triangle upper X upper Y upper Z is a dilation of triangle upper X prime

Answer 1

Look at the coordinates: X(−3,−3), Y(−3,−6), Z(−6,−3) and X′(−1,−1), Y′(−1,−2), Z′(−2,−1). None of these six points equals (0,0). Therefore the triangle and its image do not pass through the origin.

(Equivalently: X′ = (−3·1/3, −3·1/3) = (−1,−1), etc., shows the dilation is by scale factor 1/3 about the origin, and since no vertex is (0,0) the dilated triangle does not pass through the center.)