Question

1. which rigid motion would be used to prove demonstrates between two mirror - image triangles? a. rotation b. translation c. reflection d. shearing 29. which transformations do not alter the points coordinates? a. rotation b. dilation c. translation d. reflection across y = x (if point is on the line) 30. a ladder leans against a wall, forming a right triangle. the ladder is 13 ft long and the base is 5 ft from the wall. what is the height of the ladder on the wall? a. 15 ft b. 12 ft c. 10 ft d. 7 ft

Explanation:

Step1: Recall mirror - image concept

Mirror - image triangles are related by reflection. Reflection is a rigid motion that flips an object over a line of reflection, creating a mirror - like image. So for the first question about mirror - image triangles, reflection is the correct rigid motion.

Step2: Analyze transformation properties

If a point lies on the line \(y = x\), a reflection across the line \(y=x\) does not change the coordinates of the point. Let the point be \((a,a)\). When reflected across \(y = x\), using the rule \((x,y)\to(y,x)\), for the point \((a,a)\) it remains \((a,a)\). Rotation, dilation, and general translation change the coordinates of points in most cases.

Step3: Apply Pythagorean theorem

In a right - triangle formed by a ladder leaning against a wall, the length of the ladder is the hypotenuse \(c = 13\) ft and the base distance from the wall is \(a = 5\) ft. According to the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(b\) is the height on the wall. So \(b=\sqrt{c^{2}-a^{2}}=\sqrt{13^{2}-5^{2}}=\sqrt{169 - 25}=\sqrt{144}=12\) ft.

Answer:

1. C. Reflection
2. d. Reflection across \(y = x\) (if point is on the line)
3. b. 12 ft