One side of a triangle is x inches longer than another side. The ray bisecting the angle formed by these sides divides the opposite side into 3-inch and 5-inch segments. Find the perimeter of the triangle in terms of x. ( triangle angle bisector theorem)
Let y be the small length.
y/3=x+y/5
y=1.5x
Perimeter: y+x+y+8
1.5x+x+1.5x+8=4x+8
Answer is x+ 2, correct?????????!
That is correct in terms of x!
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To solve this problem, we need to follow these steps:
1. Let's assign variables to the different lengths in the triangle. We'll call the shorter side length "y" and the longer side length "y + x" since it is x inches longer than the shorter side.
2. According to the triangle angle bisector theorem, the ray bisecting the angle formed by the sides divides the opposite side into segments that have proportional lengths. In this case, we have two segments with lengths 3 inches and 5 inches.
So, we can set up the following equation: y/3 = (y + x)/5.
3. Now, we can solve this equation to find the value of y. By cross-multiplying, we get 5y = 3(y + x).
Expanding, we get 5y = 3y + 3x.
Simplifying further, we get 2y = 3x, or y = (3/2)x.
4. We now have the value of y in terms of x. To find the perimeter of the triangle, we need to add up the lengths of all three sides.
Perimeter = y + (y + x) + 3 + 5.
Perimeter = (3/2)x + (3/2)x + x + 8.
Perimeter = 3x/2 + 3x/2 + x + 8.
Perimeter = (3x + 3x + 2x + 16)/2.
Perimeter = (8x + 16)/2.
Perimeter = 4(x + 2).
So, the perimeter of the triangle, in terms of x, is 4(x + 2).