Answer:
AB=2.775
BC=5.55
CA=6.475
Step-by-step explanation:
Since midpoints split their sides in half, we can see that the triangle MNK formed by the midpoints will be half the perimeter of the triangle ABC. Since P of MNK = 7.4, we know that the perimeter of ABC = 7.4 * 2, which is 14.8. Now we can split the 14.8 so that it follows the ratio.
3+6+7=16
14.8/16=0.925
AB=0.925*3=2.775
BC=0.925*6=5.55
CA=0.925*7=6.475
The lengths of the sides of △ABC are AB=2.775 in., BC=5.55 in. and CA=6.475 in.
Given that, AB:BC:CA=3:6:7 and M, N, and K are the midpoints of the sides.
Let us take AB=3x, BC=6x and CA=7x.
The midpoint theorem states that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and equal to half of the length of the third side.
So, NK=1/2 AB=1/2 (3x), MN=1/2 BC=1/2 (6x) and MK=1/2 (AC)=1/2 (7x)
The perimeter of △MNK equals 7.4.
Now,MN+NK+MK=3x+1.5x+3.5x=7.4 in.
⇒8x=7.4 in.
⇒x=0.925 in.
So, AB=2.775 in., BC=5.55 in. and CA=6.475 in.
Therefore, the lengths of the sides of △ABC are AB=2.775 in., BC=5.55 in. and CA=6.475 in.
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