Question

if a rectangle is translated 3 units up and 4 units right, what happens to the coordinates of its vertices?
a. they are divided by 3 and 4.
b. they are scaled by 3 and 4.
c. they remain unchanged.
d. they are increased by 3 in the y-direction and 4 in the x-direction.

if two triangles have corresponding angles that are equal, what can be said about the triangles?
a. they are similar.
b. they have different shapes.
c. they are reflections.
d. they are congruent.

a triangle with side lengths 5, 12, and 13 is dilated to create a similar triangle with side lengths 30, 72, and 78. what is the dilation factor?
a. 6
b. \\(\frac{1}{2}\\)
c. 10
d. \\(\frac{6}{6}\\)

Explanation:

First Question:

Translation in coordinate geometry means shifting a figure. Moving up affects the y - coordinate (increasing it by the number of units moved up) and moving right affects the x - coordinate (increasing it by the number of units moved right). Option a is incorrect as translation is not division. Option b is incorrect as scaling is a different transformation (like dilation) not translation. Option c is incorrect as the coordinates do change during translation. Option d correctly describes the translation: increasing y - coordinate by 3 (up) and x - coordinate by 4 (right).

The AA (Angle - Angle) similarity criterion states that if two triangles have corresponding angles equal, then the triangles are similar. Similar triangles have the same shape (same angles) but not necessarily the same size. Option b is incorrect as they have the same shape. Option c is incorrect as reflection is a different transformation and just having equal angles doesn't mean they are reflections. Option d is incorrect as congruent triangles have equal corresponding angles and equal corresponding sides, while similar triangles only need equal angles (sides are in proportion). So option a is correct.

Step 1: Recall the concept of dilation factor

The dilation factor \(k\) is found by dividing the length of a side of the image (dilated triangle) by the length of the corresponding side of the original triangle.

Step 2: Calculate the dilation factor

Take the first pair of corresponding sides: original side length \(a = 5\), image side length \(a'=30\). The dilation factor \(k=\frac{a'}{a}\).
Substitute the values: \(k = \frac{30}{5}=6\). We can check with other sides: \(\frac{72}{12} = 6\) and \(\frac{78}{13}=6\). So the dilation factor is 6.

Answer:

d. They are increased by 3 in the y - direction and 4 in the x - direction.

Second Question: