overline IJ is a midsegment of triangle FGH; IJ = 10; FH = 14 and GH = 17 Find the perimeter of triangle IJH
In a triangle, if a line segment is a midsegment, it divides the opposite side into two congruent segments. Therefore, IJ = JH.
Given IJ = 10 and JH = 10.
To find the perimeter of triangle IJH, we need to find the length of FH.
Since triangle IJH and triangle FGH share the same base FH, and their heights are the same, we can use the proportional property of area. The ratio of the areas of two triangles with the same height is equal to the ratio of their bases.
Area of triangle IJH/Area of triangle FGH = IJ/FH
IJ/FH = 10/14 = 5/7
Area of triangle IJH/Area of triangle FGH = 5/7
Let's denote the area of triangle FGH as A and the area of triangle IJH as 5A.
Since the height of both triangles are the same, the ratio of their bases is equal to the ratio of their heights.
FG/FH = 17/14 = 1.2142 (approximate to 4 decimal places)
Area of triangle FGH/Area of triangle IJH = 7/5
FG/FH = 1.2142 = √(7/5)
FG = √(7/5) * FH
FG = √(7/5) * 14
Let's calculate FG:
FG ≈ √(7/5) * 14 ≈ 11.82 (approximate to 2 decimal places).
Now, we can calculate the perimeter of triangle IJH:
Perimeter = IJ + JH + FH
Perimeter = 10 + 10 + 14
Perimeter = 34
Therefore, the perimeter of triangle IJH is 34 units.