Question

1. what is the distance between points (4, 3) and (7, 7)?

a. 5 units
b. 6 units
c. 3 units
d. 4 units

1. which of these is a real - world example of rotational symmetry?

a. wind turbines
b. human face
c. brick wall
d. bookshelf

1. given two triangles with two pairs of corresponding angles congruent, which of the following can you conclude?

a. the triangles are isosceles.
b. the triangles are congruent.
c. the triangles are similar.
d. the triangles are scalene.

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $(x_1,y_1)=(4,3)$ and $(x_2,y_2)=(7,7)$.

Step2: Substitute values

$d=\sqrt{(7 - 4)^2+(7 - 3)^2}=\sqrt{3^2+4^2}=\sqrt{9 + 16}=\sqrt{25}=5$.

Step3: Recall rotational - symmetry concept

Rotational symmetry is when a shape looks the same after a rotation. Wind - turbines have blades that look the same after rotation, while human faces, brick walls, and bookshelves do not have rotational symmetry in the same sense.

Step4: Recall triangle similarity and congruence rules

If two pairs of corresponding angles of two triangles are congruent, then by the AA (angle - angle) similarity criterion, the triangles are similar. We cannot conclude they are congruent without information about side - lengths, and we cannot conclude they are isosceles or scalene just from angle information.

Answer:

1. a. 5 units
2. a. Wind turbines
3. c. The triangles are similar.