Triangle F G H is drawn with a solid line and is located in quadrant 2 of a coordinate plane. Triangle J K I is drawn with a dotted line and is located in quadrant 4.

Question

Assess two figures using the definitions of transformations to determine whether the figures are similar. Is ΔFGH∼ΔIJK , and for what reason?

(1 point)
Responses

Yes. All sides are proportional.
Yes. All sides are proportional.

Yes. All angles are congruent.
Yes. All angles are congruent.

No. If point J was at (6,−6), the triangles would be similar.
No. If point upper J was at left parenthesis 4 comma negative 5 right parenthesis , the triangles would be similar.

No. If point J was at (−6,6), the triangles would be similar.

Answer 1

To determine if triangle FGH is similar to triangle IJK (denoted as ΔFGH∼ΔIJK), we need to consider two main criteria for similarity in triangles: the proportionality of corresponding sides and the congruence of corresponding angles.

1. All sides are proportional: This means that if we can establish a constant ratio between the lengths of corresponding sides of the triangles, they could be similar due to the Side-Side-Side (SSS) similarity theorem.

2. All angles are congruent: If all corresponding angles of both triangles are equal, then the triangles are similar by the Angle-Angle (AA) similarity theorem.

Assessment of the Responses:

• Yes. All sides are proportional. - This response could imply that the triangles could be similar based on proportionality alone.

• Yes. All angles are congruent. - This response highlights that if all angles of triangle FGH are equal to those of triangle IJK, then they are similar based on angle congruence.

• No. If point J was at (6,−6), the triangles would be similar. - This response suggests that a change in the position of point J affects the similarity criteria. If point J needs to be at a specific location to establish similarity, it indicates that the current positions do not satisfy the similarity criteria.

• No. If point J was at (−6,6), the triangles would be similar. - Similar to the previous response, this indicates that point J's current position does not allow for similarity.

Given that we don't have explicit information about side lengths or angle measurements from the problem, we can't draw a definitive conclusion without seeing the actual figures. However, if we were to analyze based on general definitions:

Conclusion:

Without further information suggesting that all angles are congruent or that all corresponding sides are proportional, we cannot confirm the similarity definitively.

The most appropriate conclusion based on proposed characteristics would be:

• If all sides are proven proportional or all angles congruent through additional measures, then the triangles would be similar. But from the responses provided, the best way to explain the current situation of similarities would be that:
• No. If point J was at (−6,6), the triangles would be similar. can imply that there is a positional aspect that currently affects similarity.

To summarize: Based on the options given, either an analysis of angles or a change in position of points must be clarified to conclude the similarity definitively.

Answer 2

Suppose one figure is mapped to a similar figure on the coordinate plane. The lengths of the sides of the similar figure are twice the lengths of the corresponding sides of the original figure. In a sequence of transformations from the preimage to the image, which type of transformation must occur?(1 point)
Responses

dilation
dilation

reflection
reflection

translation
translation

rotation

Answer 3

When one figure is mapped to a similar figure on the coordinate plane and the lengths of the sides of the similar figure are twice the lengths of the corresponding sides of the original figure, this indicates a change in size while maintaining the same shape. The type of transformation that results in such a scaling of dimensions is known as a dilation.

A dilation involves resizing a figure by a scale factor relative to a point (the center of dilation), without altering its shape or angles. In this case, since the sides of the similar figure are twice the lengths of the original figure, the scale factor for this dilation is 2.

Conclusion:

The correct answer is dilation.

Answer 4

If ΔPRQ has ∠P=30 and ∠R=127 and ΔFGH has ∠H=30 and ∠G=23 , what criterion has been met to prove the triangles similar?(1 point)
Responses

SSS
SSS

ASA
ASA

SAS
SAS

AA

Answer 5

To determine the similarity criteria that applies to the triangles ΔPRQ and ΔFGH, we first need to analyze the angles given for both triangles.

For triangle ΔPRQ:

• ∠P = 30°
• ∠R = 127°

To find the third angle (∠Q), we can use the fact that the sum of the angles in a triangle is 180°: \[ ∠Q = 180° - ∠P - ∠R = 180° - 30° - 127° = 23° \]

So, the angles for triangle ΔPRQ are:

• ∠P = 30°
• ∠R = 127°
• ∠Q = 23°

For triangle ΔFGH:

• ∠H = 30°
• ∠G = 23°

We can find the third angle (∠F) using the same principle: \[ ∠F = 180° - ∠H - ∠G = 180° - 30° - 23° = 127° \]

So, the angles for triangle ΔFGH are:

• ∠H = 30°
• ∠G = 23°
• ∠F = 127°

Now, we can summarize the angles for both triangles:

• ΔPRQ: ∠P = 30°, ∠R = 127°, ∠Q = 23°
• ΔFGH: ∠H = 30°, ∠G = 23°, ∠F = 127°

Angle Correspondence

• ∠P (ΔPRQ) = ∠H (ΔFGH) = 30°
• ∠Q (ΔPRQ) = ∠G (ΔFGH) = 23°
• ∠R (ΔPRQ) = ∠F (ΔFGH) = 127°

Conclusion

Both triangles have three angles that are the same (the corresponding angles are equal). The criterion for triangle similarity that has been met here is the Angle-Angle (AA) criterion.

Therefore, the correct response is AA.

Answer 6

A dilation creates two similar triangles, ΔMNP∼ΔTUV . To use the SSS (Side-Side-Side) Similarity Theorem, which must be true?(1 point)
Responses

MNTU=MPTV=NPUV
cap m cap n over cap t cap u is equal to cap m cap p over cap t cap v is equal to cap n cap p over cap u cap v

MNTU=NPUV and ∠N≅∠U
cap m cap n over cap t cap u is equal to cap n cap p over cap u cap v and angle cap n is congruent to angle cap u

MNTU=MPUV=NPMN
cap m cap n over cap t cap u is equal to cap m cap p over cap u cap v is equal to cap n cap p over cap m cap n

∠M≅∠V and

Answer 7

To use the SSS (Side-Side-Side) Similarity Theorem for triangles ΔMNP and ΔTUV, we need to establish that the corresponding sides of the two triangles are proportional. This means that the lengths of the sides of triangle MNP compared to triangle TUV must maintain a constant ratio.

Considering the properties of dilations and the SSS criterion, we can interpret the requirements as follows:

For SSS Similarity Theorem:

1. Corresponding Sides Are Proportional: This must be true: \[ \frac{MN}{TU} = \frac{NP}{UV} = \frac{MP}{TV} \]

Thus, the correct response involves expressing the proportional relationships of the sides.

Evaluation of the Given Responses:

1. MNTU=MPTV=NPUV - This statement is unclear as it does not express a proportionality.

2. \(\frac{MN}{TU}=\frac{MP}{TV}=\frac{NP}{UV}\) - This correctly expresses the condition required for the SSS Similarity Theorem.

3. MNTU=NPUV and ∠N≅∠U - This does not directly relate to the SSS Criterion, as it involves angles.

4. MNTU=MPUV=NPMN - This also does not properly reflect the SSS condition.

5. ∠M≅∠V and - This indicates angle congruence, which is not part of the SSS criterion but rather relates to the SAS criterion.

Conclusion:

The correct response indicating what must be true to use the SSS Similarity Theorem for triangles ΔMNP and ΔTUV is:

\(\frac{MN}{TU} = \frac{NP}{UV} = \frac{MP}{TV}\).

This response accurately reflects that for SSS similarity, the ratios of the lengths of the corresponding sides must be equal.