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.
http://en.wikipedia.org/wiki/Angle_bisector_theorem
Then S/(S-x)= 5/3
and 3X=5S-5X
or 5x=3S or x=3s/5 or S=5x/3
Perimeter: (3+5)+ S+(S-x)=
8+2*5x/3 -x=
8+x(10/3-1) and you can surely simplify that.
check my thinking
Thought this question looked familiar, so I went back to find it and found your objection to my solution
http://www.jiskha.com/display.cgi?id=1326173456
Of course you were right to spot my error, it should have said
1.5x + 1.5x + x + 8
= 4x+8
Also just noticed a typo in bobpursley's solution
2nd line should have been
3S = 5S - 5X , then
5x = 2s
s = 5x/2 or 2.5x
perimeter = 8 + s + s-x
= 8 + 5x-x = 4x+8
Let's break down the information provided and solve the problem step-by-step.
Step 1: Define the variables
Let's define the variables:
- x: the length added to one side of the triangle (in inches).
Step 2: Express the lengths of the sides
Since one side of the triangle is x inches longer than the other side, we can express the lengths of the sides as:
- First side: x inches
- Second side: (x + x) = 2x inches
- Third side: ?
Step 3: Use the angle bisector to find the length of the third side
The ray bisecting the angle formed by the first and second sides divides the opposite side into two segments: 3 inches and 5 inches.
The length of the entire third side is the sum of these two segments:
Third side = 3 inches + 5 inches = 8 inches
Step 4: Calculate the perimeter of the triangle
The perimeter of a triangle is the sum of the lengths of all its sides.
Perimeter = First side + Second side + Third side
Perimeter = x inches + 2x inches + 8 inches
Step 5: Simplify the expression
Combining like terms, we can simplify the expression:
Perimeter = 3x inches + 8 inches
Therefore, the perimeter of the triangle in terms of x is 3x + 8 inches.
To find the perimeter of the triangle in terms of x, we need to determine the lengths of all three sides first.
Let's denote the lengths of the three sides of the triangle as follows:
- The length of the side adjacent to the 3-inch segment as a inches.
- The length of the side adjacent to the 5-inch segment as b inches.
- The length of the hypotenuse as c inches.
According to the information provided, one side of the triangle (b) is x inches longer than another side (a). Therefore, we can express b in terms of a as: b = a + x.
The ray bisecting the angle formed by sides a and b divides the opposite side c into two segments, measuring 3 inches and 5 inches. This means that c can be expressed as the sum of these two segments: c = 3 + 5 = 8 inches.
Using the Pythagorean theorem, we can relate the lengths of the three sides:
\(a^2 + (a + x)^2 = 8^2\)
Expanding the equation, we get:
\(a^2 + (a^2 + 2ax + x^2) = 64\)
Combining like terms:
\(2a^2 + 2ax + x^2 = 64\)
Dividing by 2 to simplify the equation:
\(a^2 + ax + \frac{x^2}{2} = 32\)
Now, we can find the value of a by solving this quadratic equation. Once we have a, we can substitute it back into the equation b = a + x to find b. Finally, we can calculate the perimeter by adding the lengths of all three sides (a + b + c) in terms of x.
Therefore, to find the perimeter of the triangle in terms of x, we need to solve the quadratic equation and substitute the values of a and b into the perimeter formula.