Triangles AED and BEC are formed using the straight lines AB and CD as shown in the figure above. If BE = BC, DE2 AD2 + AE2 and the measure of AED is x, which of the following statements must be true? CE2 2BE2 AE AD DE CEa)I onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'A'. Can you explain this answer?

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Triangles AED and BEC are formed using the straight lines AB and CD as shown in the figure above. If BE = BC, DE² > AD² + AE² and the measure of ∠AED is x, which of the following statements must be true?

CE² > 2BE²
AE < AD
DE < CE

a)
I only
b)
II only
c)
III only
d)
I, II and III
e)
None of the above

Correct answer is option 'A'. Can you explain this answer?

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Triangles AED and BEC are formed using the straight lines AB and CD as...

Given:

Triangles AED and BEC are formed using the straight lines AB and CD
- So, ∠AED = ∠CEB = x (Vertically opposite angles)

In ΔBEC,
- BE = BC
- So, ∠CEB = ∠ECB = x
- Therefore, ∠EBC = 180^∘−2x

In ΔAED,
- ∠EAD = 180^∘−(x+2x)=180^∘−3x

We are given that DE² > AD² + AE²
- This means, ΔAED is obtuse
  - So, ∠EAD > 90^∘
  - 180^∘−3x>90^∘
  - 90^∘>3x
  - That is, x<30∘

To find: Which of the 3 statements must be true?

Approach:

Based on the given and inferred information above, we’ll evaluate the 3 statements one by one

Working Out:

Evaluating Statement I
- Statement I says that CE² > 2BE²
  - That is, CE² > BE² + BC² (since we are given that BE = BC)
- This inequality will be true if ∠EBC is obtuse.
- Checking if ∠EBC is obtuse
  - ∠EBC = 180^∘−2x
  - Inferred above: x<30^∘
  - So,−2x>−60^∘ (multiplying both sides with a negative number reverses the sign of inequality)
  - 180^∘–2x>180^∘–60^∘
  - 180^∘–2x>120^∘
  - That is, ∠EBC>120^∘

Since ∠EBC is indeed obtuse, Statement I will be true for all values of x

Evaluating Statement II
- Statement II says that AE < AD
- Checking if this statement is true:
  - In ΔAED, the angles in increasing order of magnitude are:
    - ∠AED < ∠ADE < ∠DAE
      - Note that since we’ve already inferred that ∠DAE is an obtuse angle, it will definitely be the greatest angle of this triangle. Among the remaining 2 angles, the angle that measures 2x will obviously be greater than the angle that measures x.

So, the sides of this triangle in increasing order of magnitude will be:
- AD < AE < DE
Thus, Statement II is NOT true.

Evaluating Statement III
- Statement III says that DE < CE
- This inequality compares the sides of two different triangles. We do not have enough information to make this comparison. So, it is not a must be true statement based on the limited information that we have.

Getting to the final answer
- Thus, only Statement I is a must be true statement.

Looking at the answer choices, we see that the correct answer is Option A

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Triangles AED and BEC are formed using the straight lines AB and CD as shown in the figure above. If BE = BC, DE2 > AD2 + AE2 and the measure of ∠AED is x, which of the following statements must be true? CE2 > 2BE2 AE < AD DE < CEa)I onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'A'. Can you explain this answer?

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