A line parallel to one side of a triangle divides one of the other sides into segments of lengths 15 and 9 and divides the third side so that the longest portion is 36. How long is the shorter portion of the third side?
according to your description
x/36 = 9/15
x = 9(36)/15 = 21.6
To find the length of the shorter portion of the third side, we can use the concept of similar triangles.
Here's how you can solve this problem step by step:
Step 1: Draw a triangle.
Draw a triangle on a piece of paper or visualize it in your mind. Label the three sides as a, b, and c.
Step 2: Identify the line parallel to one side.
Identify the line that is parallel to one side of the triangle. Let's call this line "d."
Step 3: Identify the segments on the other side.
From the information given, we know that line d divides one of the other sides (side b) into two segments of lengths 15 and 9. Let's call these segments "x" and "y," respectively. So we have x + y = 15 (equation 1).
Step 4: Identify the longest portion of the third side.
We are also given that line d divides the third side (side c) in such a way that the longest portion is 36. Let's call the shorter portion "z." So we have z + 36 = c (equation 2).
Step 5: Establish proportions.
Since we have identified similar triangles, we can establish proportions between corresponding sides.
In the smaller triangle: (x + z) / y = a / b
In the larger triangle: 15 / 9 = a / c
Step 6: Manipulate the equations.
Rearrange equation 1 to find one variable in terms of the other:
x = 15 - y
Substitute the value of x in the proportion for the smaller triangle:
((15 - y) + z) / y = a / b
Simplify:
(15 + z - y) / y = a / b
Now, substitute the value of c from equation 2 in the proportion for the larger triangle:
15 / 9 = a / (z + 36)
Simplify:
5/3 = a / (z + 36)
Step 7: Solve the equations.
Since two proportions are equal to each other, we can set the two right-hand sides equal to each other and solve for z:
(15 + z - y) / y = 5/3
Cross-multiply:
3(15 + z - y) = 5y
Simplify:
45 + 3z - 3y = 5y
Combine like terms:
3z - 8y = -45
Now, substitute the value of y in terms of z from equation 1:
3z - 8(15 - y) = -45
Simplify:
3z - 120 + 8y = -45
3z + 8y = 75
Rearrange and solve for z:
3z = 75 - 8y
z = (75 - 8y) / 3
Step 8: Substitute the values.
To find the length of the shorter portion of the third side, we need to substitute the values of y into the equation for z.
From equation 1: x + y = 15
x + 15 - x = 15 (since x = 15 - y)
15 = 15
Therefore, we can substitute x = 0 into the equation for z:
z = (75 - 8(0)) / 3
z = 75 / 3
z = 25
So, the length of the shorter portion of the third side is 25.