Two triangles are similar. The lengths of the sides of the smaller triangle are 5,6,and 9. The perimeter of the larger triangle is 50. What are the lengths of the sides of the larger triangle?
To find the lengths of the sides of the larger triangle, we can use the fact that corresponding sides of similar triangles are proportional.
Let's denote the lengths of the corresponding sides of the larger triangle as a, b, and c.
Since the lengths of corresponding sides are proportional, we can set up the following proportions:
a/5 = b/6 = c/9
To find the lengths of the sides of the larger triangle, we also know that the perimeter of the larger triangle is 50. Therefore, we have:
a + b + c = 50
To solve these equations, we can use the knowledge that the sum of the ratios in a proportion is equal to 1:
(a/5) + (b/6) + (c/9) = 1
Multiplying through by the common denominator of 90, we get:
18a + 15b + 10c = 90
Now we have a system of equations:
a + b + c = 50
18a + 15b + 10c = 90
Simplifying the first equation, we get:
c = 50 - a - b
Substituting this into the second equation, we get:
18a + 15b + 10(50 - a - b) = 90
Simplifying this equation, we get:
18a + 15b + 500 - 10a - 10b = 90
8a + 5b = -410
Now we can solve this system of two equations. However, we need one more equation to determine the specific values of a, b, and c.
Please provide an additional equation or information to solve for the lengths of the sides of the larger triangle.
To solve this problem, we need to use the concept of similarity between triangles. According to the property of similar triangles, corresponding sides of similar triangles are in proportion.
Let's denote the sides of the larger triangle as a, b, and c. We know the perimeter of the larger triangle is 50, so we can write the equation:
a + b + c = 50
Now, let's set up proportions between corresponding sides of the two triangles:
a / 5 = b / 6 = c / 9
To solve for the lengths of the sides of the larger triangle, we need to find the value of the common ratio in the proportions.
First, let's solve for the value of a:
a / 5 = b / 6
Cross-multiplying, we get:
6a = 5b
Simplifying, we have:
a = (5/6) * b
Next, let's solve for the value of c:
c / 9 = b / 6
Cross-multiplying, we get:
6c = 9b
Simplifying, we have:
c = (3/2) * b
Now, substitute these values of a and c in the perimeter equation:
(5/6) * b + b + (3/2) * b = 50
(5/6 + 1 + 3/2) * b = 50
Multiply 6 on both sides to get rid of the denominator:
(5 + 6 + 9/2) * b = 300
(10 + 12 + 9) * b / 2 = 300
31 * b / 2 = 300
Now, solve for b:
b = (300 * 2) / 31
b ≈ 19.3548
Now, substitute the value of b back into the equations for a and c:
a = (5/6) * 19.3548
a ≈ 16.129
c = (3/2) * 19.3548
c ≈ 29.032
So, the lengths of the sides of the larger triangle are approximately 16.129, 19.3548, and 29.032.
Let the sides of the larger be 5x, 6x, and 9x
5x+6x+9x = 60
solve for x, sub x into the definitions.