Question

1. what happens to the corresponding sides of two similar triangles?

a. they are equal in length.
b. they have the same ratio.
c. they are proportional to each other.
d. they form a 90° angle.

1. if a rectangle is translated 3 units up and 4 units right, what happens to the coordinates of its vertices?

a. they are scaled by 3 and 4.
b. they are increased by 3 in the y-direction and 4 in the x-direction.
c. they are divided by 3 and 4.
d. they remain unchanged.

1. what is the length of the hypotenuse of a right triangle with legs measuring 7 units and 24 units?

a. 25 units
b. 28 units
c. 27 units
d. 26 units

Explanation:

Question 10

To determine the correct option, recall the definition of similar triangles. Similar triangles have corresponding angles equal and corresponding sides proportional (i.e., their ratios are equal). Option a is incorrect because only congruent triangles have equal - length corresponding sides. Option b is less precise than option c; "proportional" is the standard term to describe the relationship of corresponding sides in similar triangles (proportional means their ratios are equal). Option d is incorrect as there's no requirement for corresponding sides to form a 90° angle.

When a figure is translated in the coordinate plane, a translation of \(n\) units up affects the \(y\) - coordinate (increases it by \(n\)) and a translation of \(m\) units right affects the \(x\) - coordinate (increases it by \(m\)). Scaling (option a) is a different transformation, division (option c) is not related to translation, and the coordinates do not remain unchanged (option d) during a translation. So when a rectangle is translated 3 units up and 4 units right, the \(y\) - coordinates of the vertices increase by 3 and the \(x\) - coordinates increase by 4.

Step 1: Recall the Pythagorean theorem

For a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\), the Pythagorean theorem is given by \(c=\sqrt{a^{2}+b^{2}}\). Here, \(a = 7\) units and \(b=24\) units.

Step 2: Calculate \(a^{2}+b^{2}\)

First, calculate \(a^{2}=7^{2} = 49\) and \(b^{2}=24^{2}=576\). Then, \(a^{2}+b^{2}=49 + 576=625\).

Step 3: Calculate the hypotenuse \(c\)

Take the square root of 625: \(c=\sqrt{625}=25\) units.

Answer:

c. They are proportional to each other.

Question 11