Question 349259: __
in triangle ABC, CD is both the median and the altitude if AB= 5x+3, AC=2x+8 and BC = 3x+5 what is the perimeter of triangle ABC? how do u solve it
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! (Note: It will help you understand the following if you have (or make) a drawing of the problem.)
- If segment CD is an altitude then angles ADC and BDC are right angles. (Altitudes are, by definition, perpendicular to the base. And perpendicular, by definition, means right angles are formed at the intersection.)
- If segment CD is a median then segments AD and DB are congruent because, by definition, one endpoint of a median is the midpoint of a side.
- Since
- segments AD and DB are congruent
- angles ADC and BDC are right angles (which are always congruent)
- segment CD is congruent to itself
then by SAS triangles ADC and BDC are congruent. - Since triangles ADC and BDC are congruent, then segments AC and BC are congruent because they are corresponding sides of congruent triangles.
As a result of the logic above we now know that AC = BC. So
2x + 8 = 3x + 5
We can solve this for x with a little Algebra. Subtract 2x from each side:
8 = x + 5
Subtract 3 from each side:
3 = x
With this value for x we can find the lengths of the three sides of triangle ABC:
AB = 5x + 3 = 5(3) + 3 = 15 + 3 = 18
AC = 2x + 8 = 2(3) + 8 = 6 + 8 = 14
BC = 3x + 5 = 3(3) + 5 = 9 + 5 = 14
This makes the perimeter 18+14+14 = 46.
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