Question

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?

Answer 1

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.

Answer 2

if Δ ABC ≅ CDA by the HL theorem and ¯AC = 84 m and ¯AD = 85m, how long is ¯BA?

Answer 3

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.

Answer 4

if Δ ABC ≅ CDA by the HL theorem and ¯AC = 84 m and ¯AD = 85m, how long is ¯BA?

22m
13m
27m
15m

Answer 5

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.