SOLUTION: given: triangle abc is scalene altitudes ae,cd prove: ae is not congruent to cd

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Question 1177132: given: triangle abc is scalene
altitudes ae,cd
prove: ae is not congruent to cd
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source): You can put this solution on YOUR website!
**1. Given Information**
* Triangle ABC is scalene. This means that all three sides of the triangle have different lengths.
* AE and CD are altitudes of triangle ABC. This means that AE is perpendicular to BC, and CD is perpendicular to AB.
**2. Assumption**
Let's assume, for the sake of contradiction, that AE is congruent to CD.
**3. Area of a Triangle**
The area of a triangle can be calculated as (1/2) * base * height.
**4. Applying the Area Formula**
* Area of triangle ABC using base BC and altitude AE: (1/2) * BC * AE
* Area of triangle ABC using base AB and altitude CD: (1/2) * AB * CD
**5. Using the Assumption**
Since we assumed AE ≅ CD, we can substitute AE for CD in the area formulas:
* (1/2) * BC * AE = (1/2) * AB * AE
**6. Simplifying the Equation**
If we divide both sides of the equation by (1/2) * AE, we get:
* BC = AB
**7. Contradiction**
This result contradicts our given information that triangle ABC is scalene (meaning all sides have different lengths).
**8. Conclusion**
Therefore, our initial assumption that AE ≅ CD must be false. Hence, we have proven that AE is not congruent to CD.
Answer by ikleyn(52798)   (Show Source): You can put this solution on YOUR website!
.
given: triangle ABC is scalene
altitudes AE, CD
prove: AE is not congruent to CD
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

The fact that triangle ABC is a scalene means that there is no congruent pair of sides.


The area of triangle ABC is

    area = (1/2)*|BC|*|AE| = (1/2)*|AB|*|CD|.    (1)


Assume for a minute that AE is congruent to CD.

Then from equality (1), by canceling equal factors (1/2), |AE| and |CD|, we get

    |BC| = |AB|.


But it contradicts to the given fact that triangle ABC is scalene.


The contradiction proves that AE is not congruent to CD.

Solved.



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