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Assess two figures using the definitions of transformation to determine whether the figures are similar. Is Delta FGH/sim/Delta IJK, and for what reason?

F is at -5,5 G -2,3 H -5,1
I is at 9,-8 J 4,5 K 9,-4

11 answers

To determine if the triangles ΔFGH and ΔIJK are similar, we need to assess the corresponding angles and the lengths of the sides to see if they maintain the same ratios. Two triangles are similar if:

  1. Their corresponding angles are equal (Angle-Angle or AA similarity).
  2. The lengths of corresponding sides are proportional (Side-Side-Side or SSS similarity).

Step 1: Calculate the lengths of the sides of both triangles.

For Triangle FGH:

  • FG:
    \[ FG = \sqrt{(-2 - (-5))^2 + (3 - 5)^2} = \sqrt{(-2 + 5)^2 + (3 - 5)^2} = \sqrt{(3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \]

  • GH:
    \[ GH = \sqrt{(-5 - (-2))^2 + (1 - 3)^2} = \sqrt{(-5 + 2)^2 + (1 - 3)^2} = \sqrt{(-3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \]

  • FH:
    \[ FH = \sqrt{(-5 - (-5))^2 + (1 - 5)^2} = \sqrt{0^2 + (-4)^2} = \sqrt{0 + 16} = 4 \]

For Triangle IJK:

  • IJ:
    \[ IJ = \sqrt{(4 - 9)^2 + (5 - (-8))^2} = \sqrt{(-5)^2 + (5 + 8)^2} = \sqrt{25 + 169} = \sqrt{194} \]

  • JK:
    \[ JK = \sqrt{(9 - 4)^2 + (-4 - 5)^2} = \sqrt{(5)^2 + (-9)^2} = \sqrt{25 + 81} = \sqrt{106} \]

  • IK:
    \[ IK = \sqrt{(9 - 9)^2 + (-4 - (-8))^2} = \sqrt{0^2 + (4)^2} = \sqrt{16} = 4 \]

Step 2: Compare the ratios of the sides.

  • For Triangle FGH:

    • Side lengths are: \( \sqrt{13}, \sqrt{13}, 4 \) (we can denote them as \( a, a, b \))

  • For Triangle IJK:

    • Side lengths are: \( \sqrt{194}, \sqrt{106}, 4 \) (we can denote them as \( c, d, b \))

Check the proportionality:

Since we have a side of length 4 in both triangles, we can use that as a reference.

  1. Comparing the other two sides:
    • Using the side of length 4 to find the ratio: \[ \frac{FG}{IK} = \frac{\sqrt{13}}{4} \] \[ \frac{GH}{IK} = \frac{\sqrt{13}}{4} \] \[ \frac{IJ}{b} = \frac{\sqrt{194}}{4} \] \[ \frac{JK}{b} = \frac{\sqrt{106}}{4} \]

Conclusion:

The ratios calculated do not yield a consistent similarity ratio, as sides \( FG \) and \( GH \) are both of length \( \sqrt{13} \) while \( IJ \) and \( JK \) do not maintain overall consistent ratios when compared.

Since the angles cannot be determined directly with the information provided, it would depend on further geometric methods either through angle measurements or more coordinates analysis to check the triangles' proportionality.

In conclusion, the triangles ΔFGH and ΔIJK are not similar as their corresponding sides do not maintain proportionality while their respective angles have not been directly shown to be equal either.

If point J was at left 6, -6 right would the triangles be similar?

To determine if the triangles ΔFGH and ΔIJK are similar after updating the coordinates of point J to (3, -6), we will follow a similar procedure as before: calculate the lengths of the sides of both triangles and check their proportionality.

Updated Coordinates:

  • F: (-5, 5)
  • G: (-2, 3)
  • H: (-5, 1)
  • I: (9, -8)
  • J: (3, -6) 🡨 updated point
  • K: (9, -4)

Step 1: Calculate the lengths of the sides of the triangles.

For Triangle FGH:

  • FG: \[ FG = \sqrt{(-2 - (-5))^2 + (3 - 5)^2} = \sqrt{(3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \]

  • GH: \[ GH = \sqrt{(-5 - (-2))^2 + (1 - 3)^2} = \sqrt{(-3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \]

  • FH: \[ FH = \sqrt{(-5 - (-5))^2 + (1 - 5)^2} = \sqrt{0^2 + (-4)^2} = \sqrt{0 + 16} = 4 \]

For Triangle IJK:

  • IJ: \[ IJ = \sqrt{(3 - 9)^2 + (-6 - (-8))^2} = \sqrt{(-6)^2 + (2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \]

  • JK: \[ JK = \sqrt{(9 - 3)^2 + (-4 - (-6))^2} = \sqrt{(6)^2 + (2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \]

  • IK: \[ IK = \sqrt{(9 - 9)^2 + (-4 - (-8))^2} = \sqrt{0^2 + (4)^2} = \sqrt{16} = 4 \]

Step 2: Compare the ratios of the sides.

For Triangle FGH:

  • Side lengths are: \( \sqrt{13}, \sqrt{13}, 4 \)

For Triangle IJK:

  • Side lengths are: \( 2\sqrt{10}, 2\sqrt{10}, 4 \)

Ratios:

Using the side of length \( 4 \) as a reference:

  1. Comparing the remaining sides:

  • For Triangle FGH:

    • Both remaining sides are \( \sqrt{13} \).

  • For Triangle IJK:

    • Both remaining sides are \( 2\sqrt{10} \).

Setting up the ratios based on the side of length \( 4 \): \[ \frac{FG}{IK} = \frac{\sqrt{13}}{4} \] \[ \frac{GH}{JK} = \frac{\sqrt{13}}{4} \] \[ \frac{IJ}{FH} = \frac{2\sqrt{10}}{4} = \frac{\sqrt{10}}{2} \]

Similarity Check:

For the triangles to be similar, the ratios must be consistent. To check for similarity:

  1. \( \frac{\sqrt{13}}{4} \) (for FG and IK)
  2. \( \frac{\sqrt{13}}{4} \) (for GH and JK)
  3. \( \frac{\sqrt{10}}{2} \) (for IJ and FH)

Conclusion:

Since the ratios \( \frac{\sqrt{13}}{4} \) do not equal \( \frac{\sqrt{10}}{2} \) (which simplifies to \( \frac{5}{5} \) when roots and multiplication are applied), the triangles ΔFGH and ΔIJK are not similar.

Thus, even after the change to point J’s coordinates, the two triangles do not satisfy the conditions for similarity.

Are the angels congruent at all or are all the sides proportional?

To assess whether the triangles ΔFGH and ΔIJK (with the updated coordinates for J) are similar, we will determine if their corresponding angles are congruent or their sides are proportional.

Updated Coordinates:

  • F: (-5, 5)
  • G: (-2, 3)
  • H: (-5, 1)
  • I: (9, -8)
  • J: (3, -6)
  • K: (9, -4)

Step 1: Calculate the Lengths of the Sides

We already calculated the side lengths in the previous analysis.

For Triangle FGH:

  • \(FG = \sqrt{13}\)
  • \(GH = \sqrt{13}\)
  • \(FH = 4\)

For Triangle IJK:

  • \(IJ = 2\sqrt{10}\)
  • \(JK = 2\sqrt{10}\)
  • \(IK = 4\)

Step 2: Side Proportionality Check

To check for proportionality, we need to compare the lengths of the respective sides:

  1. For \(FH\) and \(IK\):

    • \(FH = 4\)
    • \(IK = 4\)

    \[ \frac{FH}{IK} = \frac{4}{4} = 1 \]

  2. For \(FG\) and \(IJ\):

    • \(FG = \sqrt{13}\)
    • \(IJ = 2\sqrt{10}\)

    To compare the side lengths, we can express both sides under a common reference to see if they equal:

    To check if \(\frac{FG}{IJ}\) holds true: \[ \frac{\sqrt{13}}{2\sqrt{10}} = \frac{\sqrt{13}}{\sqrt{40}} = \frac{\sqrt{13}}{\sqrt{40}} = \sqrt{\frac{13}{40}} \]

  3. For \(GH\) and \(JK\):

    • \(GH = \sqrt{13}\)
    • \(JK = 2\sqrt{10}\)

    Again, we apply the same method: \[ \frac{GH}{JK} = \frac{\sqrt{13}}{2\sqrt{10}} = \sqrt{\frac{13}{40}} \]

Step 3: Conclusion on Proportionality

From the results, we can conclude:

  • Sides (FK, GH) to IJ and JK are not proportional since \(\sqrt{\frac{13}{40}}\) does not equal \(1\).

Step 4: Is There Congruent Angles?

To check if the angles are congruent, we can use the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles. We can compute the angles of both triangles:

For Triangle FGH:

Using sides \(a = \sqrt{13}, b = \sqrt{13}, c = 4\):

\[ \cos(\angle FGH) = \frac{a^2 + b^2 - c^2}{2ab} = \frac{(\sqrt{13})^2 + (\sqrt{13})^2 - 4^2}{2(\sqrt{13})(\sqrt{13})} \] \[ = \frac{13 + 13 - 16}{2(\sqrt{13})(\sqrt{13})} = \frac{10}{26} = \frac{5}{13} \]

For Triangle IJK:

Using sides \(a = 2\sqrt{10}, b = 2\sqrt{10}, c = 4\): \[ \cos(\angle IJK) = \frac{a^2 + b^2 - c^2}{2ab} = \frac{(2\sqrt{10})^2 + (2\sqrt{10})^2 - 4^2}{2(2\sqrt{10})(2\sqrt{10})} \] \[ = \frac{40 + 40 - 16}{8 \cdot 10} = \frac{64}{80} = \frac{4}{5} \]

Summary:

  • The sides are not all proportional.
  • The angles found are also not equal since \( \frac{5}{13} \neq \frac{4}{5} \).

Thus, the triangles ΔFGH and ΔIJK are neither similar nor do they have congruent angles.

Thank you

You're welcome! If you have any more questions or need further assistance, feel free to ask. Have a great day!

I'm sorry, I gave you the wrong measurement for J , it should be 4,-5 not 3, -6

No problem! Let’s re-evaluate the triangles ΔFGH and ΔIJK with the corrected coordinates for point J at (4, -5).

Updated Coordinates:

  • F: (-5, 5)
  • G: (-2, 3)
  • H: (-5, 1)
  • I: (9, -8)
  • J: (4, -5) 🡨 updated point
  • K: (9, -4)

Step 1: Calculate the Lengths of the Sides

For Triangle FGH:

  • FG: \[ FG = \sqrt{(-2 - (-5))^2 + (3 - 5)^2} = \sqrt{(3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \]

  • GH: \[ GH = \sqrt{(-5 - (-2))^2 + (1 - 3)^2} = \sqrt{(-3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \]

  • FH: \[ FH = \sqrt{(-5 - (-5))^2 + (1 - 5)^2} = \sqrt{0^2 + (-4)^2} = \sqrt{0 + 16} = 4 \]

For Triangle IJK:

  • IJ: \[ IJ = \sqrt{(4 - 9)^2 + (-5 - (-8))^2} = \sqrt{(-5)^2 + (3)^2} = \sqrt{25 + 9} = \sqrt{34} \]

  • JK: \[ JK = \sqrt{(9 - 4)^2 + (-4 - (-5))^2} = \sqrt{(5)^2 + (1)^2} = \sqrt{25 + 1} = \sqrt{26} \]

  • IK: \[ IK = \sqrt{(9 - 9)^2 + (-4 - (-8))^2} = \sqrt{0^2 + (4)^2} = \sqrt{16} = 4 \]

Step 2: List Side Lengths

  • Triangle FGH:

    • \( FG = \sqrt{13} \)
    • \( GH = \sqrt{13} \)
    • \( FH = 4 \)

  • Triangle IJK:

    • \( IJ = \sqrt{34} \)
    • \( JK = \sqrt{26} \)
    • \( IK = 4 \)

Step 3: Check for Side Proportionality

To compare the corresponding sides:

  1. For \( FH \) and \( IK \):

    • \( FH = 4 \)
    • \( IK = 4 \)

    \[ \frac{FH}{IK} = \frac{4}{4} = 1 \]

  2. For \( FG \) and \( IJ \):

    • \( FG = \sqrt{13} \)
    • \( IJ = \sqrt{34} \)

    \[ \frac{FG}{IJ} = \frac{\sqrt{13}}{\sqrt{34}} \]

  3. For \( GH \) and \( JK \):

    • \( GH = \sqrt{13} \)
    • \( JK = \sqrt{26} \)

    \[ \frac{GH}{JK} = \frac{\sqrt{13}}{\sqrt{26}} = \frac{1}{\sqrt{2}} \]

Step 4: Evaluate Proportions

  • \( \frac{FH}{IK} = 1 \)
  • \( \frac{FG}{IJ} = \frac{\sqrt{13}}{\sqrt{34}} \)
  • \( \frac{GH}{JK} = \frac{1}{\sqrt{2}} \)

The ratios do not maintain consistency across the triangles' sides. Thus:

Conclusion

  • The sides are not all proportional:
    • \( FH \) and \( IK \) are equal.
    • However, \( \frac{FG}{IJ} \) and \( \frac{GH}{JK} \) do not match the ratio of \( 1 \).

Angle Congruency Check

To check for angle congruence, you could apply the Law of Cosines, but it's already clear that the sides are primarily not proportional.

Thus, the triangles ΔFGH and ΔIJK with the updated coordinates for J are neither similar nor congruent based on the lengths evaluated. If you would like to check angles using the Law of Cosines, please let me know!

Thank you again

You're welcome! If you have any more questions or need further assistance in the future, don't hesitate to reach out. Have a great day!