georgia is justifying the SSS triangle congruence criteria using △ ABC and △LMN (image 1). she transforms the first triangle with a translation to create △A'B'C' (image 2), followed by a rotation to create △ A" B" C" (image 3). she is surprised to see that △A" B " C " overlaps △LMN without having done a reflection. what happened?
Based on the given information, it seems that Georgia performed a translation followed by a rotation to transform triangle ABC into A'B'C' and then A"B"C". Triangle A"B"C" overlaps with triangle LMN without the need for a reflection.
This can happen if the translation and rotation together maintain the same orientation and shape of the triangles. In other words, if the translation of A'B'C' and rotation of A"B"C" preserve the same angles and side lengths, the resulting triangles will be congruent without requiring a reflection.
It is possible that the translation and rotation that Georgia performed were such that they resulted in the same relative positions and orientations of the vertices, angles, and side lengths of triangles A'B'C' and A"B"C" compared to triangles ABC and LMN.
This situation is not a contradiction to the SSS (Side-Side-Side) congruence criteria. The SSS criterion states that if the corresponding sides of two triangles are equal in length, then the triangles are congruent. Since Georgia's transformations maintained the same side lengths, the congruence between A"B"C" and LMN can still be justified using the SSS criterion.
if Δ ABC ≅ CDA by the HL theorem and ¯AC = 84 m and ¯AD = 85m, how long is ¯BA?
Based on the information given, we know that triangle ABC is congruent to triangle CDA by the HL (Hypotenuse-Leg) theorem. This theorem states that if a right-angled triangle has the hypotenuse and one leg congruent to the corresponding parts of another right-angled triangle, then the two triangles are congruent.
In triangle CDA, we are given that AD = 85 m.
Now, since triangle ABC is congruent to triangle CDA, we can conclude that angle B in triangle ABC is congruent to angle D in triangle CDA, and angle A in triangle ABC is congruent to angle C in triangle CDA. Therefore, triangle ABC is also a right-angled triangle.
So, in triangle ABC, we have:
- Hypotenuse AC = 84 m
- Leg AD = 85 m
We can use the Pythagorean theorem to find the length of the remaining leg, BA.
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Using the Pythagorean theorem, we have:
AC^2 = AD^2 + BA^2
Substituting the given values, we get:
84^2 = 85^2 + BA^2
7056 = 7225 + BA^2
Rearranging the equation, we have:
BA^2 = 7056 - 7225
BA^2 = -169
Since the square of a length cannot be negative, there is no real solution for BA in this scenario. Therefore, we cannot determine the length of segment BA.
if Δ ABC ≅ CDA by the HL theorem and ¯AC = 84 m and ¯AD = 85m, how long is ¯BA?
22m
13m
27m
15m
To find the length of segment BA, we can use the fact that triangle ABC is congruent to triangle CDA by the HL (Hypotenuse-Leg) theorem. This means that the hypotenuse AC in triangle ABC is congruent to the hypotenuse AD in triangle CDA, and the leg BA in triangle ABC is congruent to the leg CD in triangle CDA.
Given that ¯AC = 84 m and ¯AD = 85 m, we know that AC is congruent to AD. Therefore, ¯BA is congruent to ¯CD.
Since ¯CD is not provided in the given information, we cannot determine the exact length of ¯BA.
In this scenario, Georgia started with two triangles: △ ABC and △ LMN. She justified the SSS (Side-Side-Side) triangle congruence criteria using these triangles.
To continue her transformation, Georgia performed a translation on triangle △ ABC, creating the triangle △ A'B'C' (image 2). In a translation, the shape is moved to a different location without any changes in size or shape. This means that corresponding sides and angles of the two triangles remain congruent.
Next, Georgia performed a rotation on triangle △ A'B'C', creating the triangle △ A" B" C" (image 3). When a shape is rotated, it is turned around a fixed point. In this case, the corresponding sides and angles of the transformed and the original triangles remain congruent.
Georgia then noticed that △ A" B" C" overlaps △ LMN without having done a reflection. This means that the two triangles have congruent sides, congruent angles, and are in the same plane, resulting in an overlap or superimposition.
In short, the overlap occurred because Georgia performed a translation followed by a rotation on the initial triangle, resulting in a congruent triangle that can overlap another triangle without requiring a reflection.
Based on the given information, it seems that Georgia is justifying the SSS (Side-Side-Side) congruence criteria using two triangles △ABC and △LMN. She performs a series of transformations on △ABC, starting with a translation to create △A'B'C', and then a rotation to create △A"B"C".
It is stated that Georgia is surprised to see that △A"B"C" overlaps with △LMN without having done a reflection. In order to explain what happened, we need to understand the properties of the transformations Georgia performed.
1. Translation: A translation is a transformation that slides an object in a particular direction without changing its shape, size, or orientation. It is performed by maintaining the same direction and distance throughout the movement.
2. Rotation: A rotation is a transformation that turns an object around a fixed point, known as the center of rotation. The object is turned by a certain angle in a clockwise or counterclockwise direction.
Based on the information provided, Georgia performed a translation first to create △A'B'C'. This means that she slid △ABC in a particular direction without any rotation or reflection. Therefore, the position of the corresponding points of △A'B'C' and △ABC would have changed, but their shape and size would remain the same.
Next, Georgia performed a rotation to create △A"B"C". This means that she turned △A'B'C' around a fixed point. The result would be a rotated image of △A'B'C', maintaining the same shape and size.
Now, the surprising part is that △A"B"C" overlaps with △LMN without having done a reflection. It is important to note that a reflection, which creates a mirror image of the original object, was not specifically mentioned as being performed by Georgia.
Therefore, the most likely reason for the overlap is that the combination of translation and rotation performed by Georgia unintentionally created a reflection. This could happen if the translation and rotation together produced an effect similar to a reflection.
To determine the exact reason for the overlap, additional information or the specific properties of the translation and rotation performed by Georgia would be required.