Question 302444: the lengths of the sides of a triangle are 9 12 and 15. What is the perimeter of the triangle formed by joining the midpoints of these sides?
Answer by Nate(3500)   (Show Source):
You can put this solution on YOUR website! Draw a triangle with lengths: 9, 12 and 15.
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Highlight the midpoint of each length and connect those highlighted points to make four triangles.
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Now, we are going to solve your question using the three outer triangles ... so, disregard the innermost one.
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I will do one solution. Then, the rest will be easy to solve on your part.
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Between length 12 and 15:
We have a small triangle with lengths 7.5, 6 and unknown. The angle between 6 and 7.5 is unknown as well; however, that angle can be found using the law of cosines for the original triangle with lengths 9,12,15.
9^2 = 12^2 + 15^2 - 2*12*15*cos( angle )
The angle is about 36.9 degrees. The smaller triangle is similar to the larger, and therefore, will have the same angle degree. Now, using law of cosines again will solve the unknown side.
a^2 = 6^2 + 7.5^2 - 2*6*7.5*cos( 36.9 )
a = 4.5
The other two lengths must now be found using the same procedure in order to find the perimeter of the triangle formed by connecting the midpoints.
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