A ladder is placed so that its foot is 2 m from a wall. The ladder touches the top of a 2 m fence that is 1.5 m from the wall. State how you know the two triangles in the diagram are similar. How high up the wall does the ladder reach?

Thank you so much for your answer, Damon, i'm new to trigonometry and i'd like to know why it's necessary to do 2 - 1.5 to get 0.5? Thanks! much appreciated.

To determine whether the two triangles in the diagram are similar, we can use the property of similar triangles, which states that if the corresponding angles of two triangles are congruent and the lengths of their corresponding sides are proportional, then the triangles are similar.

In the given scenario, we have two triangles:
1. Triangle formed by the ladder, the wall, and the ground.
2. Triangle formed by the ladder, the fence, and the ground.

Now we can examine the angles and sides of the two triangles:
1. Angles: Both triangles share the same angle at the top, where the ladder touches the wall. Additionally, the angles at the base of the triangles, where the ladder meets the ground, are congruent as the ladder is a straight line.
2. Sides: The ladder is the same length in both triangles, so the ratio of their lengths is 1:1. The lengths of the other two sides for both triangles are also proportional.

Hence, based on the congruent angles and proportional side lengths, we can conclude that the two triangles are similar.

To find how high up the wall the ladder reaches, we can use the properties of similar triangles. Since the triangles are similar, the ratio of the corresponding side lengths is equal. We can set up a proportion:

(height the ladder reaches) / (length of the ladder) = (height of the fence) / (distance between the ladder and the fence)

Plugging in the given values:
(height the ladder reaches) / (2m) = (2m) / (1.5m)

Now, cross-multiplying the equation:
(height the ladder reaches) = (2m) * (2m) / (1.5m)

Simplifying the expression:
(height the ladder reaches) = 4m / 1.5

Calculating the value:
(height the ladder reaches) ≈ 2.67m or 2.7m (rounded to one decimal place)

Therefore, the ladder reaches approximately 2.7 meters up the wall.

well, I do not see your diagram but suspect you mean the two triangles under the latter which share the angle at the ladder/ground point and a right angle at the base of the fence and the wall. Therefore angle-angle -angle ---> similar

.5/2 = 2/h
h = 4/.5 = 8 meters high