Hi
Drawing a diagram will help:
A ladder 5m long leans against a wall, with the top of the ladder (B) reaching 4m up the wall and the foot of the ladder (A) is 3m from the wall (C). A man is at a point D which is 2m up the ladder.Point E lies on line AC and Point F lies on line BC. i.e. EDFC makes a square.
How can I prove that Triangle ADE is similar to triangle DBF?
Also, how can I find DE length and DF length?
Please provide working, too! Thanks so much!
To prove that triangle ADE is similar to triangle DBF, we need to show that the corresponding angles are equal and the corresponding sides are in proportion.
Step 1: Proving corresponding angles:
Angle ADE is formed between the ladder (AD) and the wall (DE).
Angle DBF is formed between the ladder (BD) and the wall (BF).
Given that ADE and DBF form a square (EDFC), it follows that angles ADE and DBF are both right angles (90 degrees).
Step 2: Proving corresponding sides:
We can use the property of similar triangles that states the ratio of the lengths of corresponding sides is equal.
Let's compare the corresponding sides:
Side AD in triangle ADE corresponds to side BD in triangle DBF (because they are both the ladders).
Side DE in triangle ADE corresponds to side BF in triangle DBF (because they are both the walls).
It can be observed that side AD is equal to side BD (as they are the same ladder).
Therefore, side DE must be equal to side BF for the corresponding sides to be in proportion.
Since corresponding angles are equal, and corresponding sides are in proportion, we can conclude that triangle ADE is similar to triangle DBF.
Now, let's find the lengths of DE and DF:
To find DE:
We can use the Pythagorean theorem to find the length of DE. The theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
DE is the hypotenuse of the right-angled triangle ADE, with sides AD = 2m and AE = 3m.
DE^2 = AD^2 + AE^2
DE^2 = (2m)^2 + (3m)^2
DE^2 = 4m^2 + 9m^2
DE^2 = 13m^2
DE = √(13m^2)
DE ≈ 3.60555m (rounded to five decimal places)
To find DF:
We can use the Pythagorean theorem again to find the length of DF. DF is the hypotenuse of the right-angled triangle DBF, with sides BD (which is equal to AD) and BF.
DF^2 = BD^2 + BF^2
DF^2 = (2m)^2 + (4m)^2 (since BF = 4m, as given)
DF^2 = 4m^2 + 16m^2
DF^2 = 20m^2
DF = √(20m^2)
DF ≈ 4.47214m (rounded to five decimal places)
Thus, the length of DE is approximately 3.60555m, and the length of DF is approximately 4.47214m.
To prove that triangle ADE is similar to triangle DBF, we need to show that their corresponding angles are congruent, and their corresponding sides are proportional.
1. Proving corresponding angles:
Angle ADE and angle DBF:
Angle ADE is the angle between the ladder and the wall, and angle DBF is the angle between the ladder and the ground. Since the ladder creates a right angle with both the wall and the ground, both of these angles are right angles and thus congruent.
Angle DAE and angle FBD:
Angle DAE is the angle between the ground and the wall, and angle FBD is the angle between the ground and the ladder. These angles are vertical angles, which are always congruent.
2. Proving corresponding sides:
Side AD and side BD:
According to the given information, point D is 2m up the ladder. This means that the length of side AD is 2m. Since both triangle ADE and triangle DBF share side AD, this side is common to both triangles, and therefore, it is congruent.
Side AE and side BF:
To find the length of side AE, we need to use the Pythagorean theorem. In triangle ADE, the hypotenuse (side AD) is 2m, and the side opposite to angle DAE (side AE) is unknown. The side adjacent to angle DAE (side DE) is given as 3m. Using the Pythagorean theorem, we have:
AD^2 = AE^2 + DE^2
2^2 = AE^2 + 3^2
4 = AE^2 + 9
AE^2 = 4 - 9
AE^2 = -5
Since the equation gives a negative value, it means that side AE is an imaginary length and does not exist. However, this does not affect the similarity between the triangles, as we can still compare their other corresponding sides.
Side DE and side DF:
Since the ladder creates a right angle with the wall, side DE is perpendicular to side EF, which is part of the ladder. Therefore, both sides DE and DF are equal in length.
Now, let's find the lengths of DE and DF:
DE = 3m (given in the problem)
DF = DE = 3m (as explained above)
To summarize:
Triangle ADE is similar to triangle DBF because their corresponding angles are congruent (both have a right angle and a pair of vertical angles) and their corresponding sides are proportional (side AD = side BD, and side DE = side DF). The lengths of DE and DF are both 3m.