The diagram shows a triangle ADE:
AE = DE
AB : BC : CD = 1 : 2 : 1
Prove that triangle ACE is congruent to triangle DBE. - Edexcel - GCSE Maths - Question 20 - 2022 - Paper 3
Question 20
The diagram shows a triangle ADE:
AE = DE
AB : BC : CD = 1 : 2 : 1
Prove that triangle ACE is congruent to triangle DBE.
Worked Solution & Example Answer:The diagram shows a triangle ADE:
AE = DE
AB : BC : CD = 1 : 2 : 1
Prove that triangle ACE is congruent to triangle DBE. - Edexcel - GCSE Maths - Question 20 - 2022 - Paper 3
Step 1
Prove angle AEC = angle DEB (isosceles triangle property)
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Answer
Since AE = DE, we can say that triangle AEC and triangle DEB are isosceles. Therefore, the angles opposite to these equal sides are equal, which gives us that angle AEC = angle DEB.
Step 2
Show that AB : BC : CD = 1 : 2 : 1 leads to proportional sides
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Answer
Since AB : BC : CD = 1 : 2 : 1, we can split segment AC into ratios such that we have AC = AB + BC + CD. Thus, the segments can be expressed as:
AB = x
BC = 2x
CD = x
This shows that AC is twice as long as AB and CD, preserving the ratios.
Step 3
Establish that AC = DE and link to congruence
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Answer
Given that AE = DE, and we have established the relationships among the segments, we can conclude that AC = DE. Therefore, by the Side-Angle-Side (SAS) criteria for triangle congruence, triangles ACE and DBE are congruent.
