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)
Reiny, if a+a+x+8 then how come it is 1.5x+x+8=2.5x+8? I thought that it is 1.5x+1.5x+x+8=4x+8. Previous problem: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.
Anonymous is correct, for the answer is 4x+8.
Well, well, well, we have ourselves a triangle with a little bit of a geometry puzzle. Let's get cracking, shall we?
We're given that one side of the triangle is x inches longer than the other side. So, let's call the shorter side "y" inches. That means the longer side is "y + x" inches. Keep up, it gets tricky!
Now, the big twist in this tale is that the angle bisector divides the opposite side into 3-inch and 5-inch segments. So, the length of the segment adjacent to the shorter side is 3 inches and the length of the segment adjacent to the longer side is 5 inches. Got it?
Alrighty then! It's time for some perimeter action. The perimeter of a triangle, as you might know, is just the sum of all its sides. So let's add up our sides, shall we?
We have y inches for the shorter side, y + x inches for the longer side, 3 inches for the adjacent segment, and 5 inches for the other adjacent segment. Add it all up and we get...
Perimeter = y + (y + x) + 3 + 5
Now, we just need to simplify that expression a bit more. Combine like terms and we end up with...
Perimeter = 2y + x + 8
And there you have it, my friend! The perimeter of this triangle, in terms of x, is 2y + x + 8. That's the final answer... for now.
To solve this problem, we can use the triangle angle bisector theorem. According to this theorem, if a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the lengths of the other two sides.
Let's call the length of the shorter side of the triangle "a" (in inches). Since one side of the triangle is x inches longer than the other, we can call the length of the longer side "a + x" (in inches).
The ray bisects the angle formed by these sides, dividing the opposite side into a 3-inch segment and a 5-inch segment. Let's call the length of the 3-inch segment "b" (in inches) and the length of the 5-inch segment "c" (in inches).
Using the triangle angle bisector theorem, we know that the ratio of the 3-inch segment to the 5-inch segment is equal to the ratio of the lengths of the shorter side to the longer side.
Therefore, we have the following equation:
b/c = a/(a + x)
Now, let's solve for the value of "a" in terms of "x":
(b/c)(a + x) = a
(a + x)b = ac
ab + xb = ac
ab = ac - xb
ab = a(c - x)
Dividing both sides of the equation by (c - x), we get:
a = (ab)/(c - x)
Now, let's find the perimeter of the triangle in terms of "x":
Perimeter = a + (a + x) + (b + c)
Since we have found the value of "a" in terms of "x", we can substitute it into the Perimeter equation:
Perimeter = [(ab)/(c - x)] + [(ab)/(c - x) + x] + (b + c)
Simplifying this equation will give us the perimeter of the triangle in terms of "x".
let the shorter side be a, then the longer side is a+x
by the Triangle Angle Bisector Theorm
(a+x)/5 = a/3
3a + 3x = 5a
3x = 2a
a = 3x/2 = 1.5x
so the sum of the three sides
= a + a+x + 8
= 1.5x + x + 8
= 2.5x + 8