I am given two triangles
One triangle has side length of 4 and 2x+7
The other one has side length of x and 1.
How do I find the lengths of the sides of these two similiar triangles
I know I use proportions but Im stuck
if the two sides given are corresponding sides, then they have a common ratio:
x/4 = 1/(2x+7)
x(2x+7) = 4
2x^2+7x-4=0
(2x-1)(x+4) = 0
x = 1/2
so the triangles have sides of
4,8 and 1/2,1
To find the lengths of the sides of two similar triangles, we can set up a proportion. In a pair of similar triangles, each corresponding side length is in proportion to its corresponding side length in the other triangle.
Let's set up the proportion using the given information:
(Length of corresponding side in triangle 1) / (Length of corresponding side in triangle 2) = (Length of another side in triangle 1) / (Length of another side in triangle 2)
For the first pair of corresponding sides:
4 / x = (2x + 7) / 1
To solve this equation, we can cross-multiply:
4 * 1 = x * (2x + 7)
Simplifying:
4 = 2x^2 + 7x
Rearranging the equation:
2x^2 + 7x - 4 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 2, b = 7, and c = -4. Substituting the values into the quadratic formula:
x = (-7 ± √(7^2 - 4 * 2 * -4)) / (2 * 2)
Simplifying further:
x = (-7 ± √(49 + 32)) / 4
x = (-7 ± √(81)) / 4
x = (-7 ± 9) / 4
Therefore, the two possible values for x are:
x₁ = (-7 + 9) / 4 = 1/2
x₂ = (-7 - 9) / 4 = -4
Now, we can find the lengths of the sides in each triangle.
For triangle 1:
- Side AB = 4
- Side BC = 2x + 7
For triangle 2:
- Side AB = x
- Side BC = 1
Substituting the values of x into the triangle 1:
For x = 1/2:
- Side AB = 4
- Side BC = 2 * (1/2) + 7 = 8/2 + 7 = 4 + 7 = 11
For x = -4:
- Side AB = 4
- Side BC = 2 * (-4) + 7 = -8 + 7 = -1
Therefore, for x = 1/2, the lengths of the sides in the two triangles are:
Triangle 1: AB = 4, BC = 11
Triangle 2: AB = 1, BC = 1
And for x = -4, the lengths of the sides in the two triangles are:
Triangle 1: AB = 4, BC = -1 (not possible as it can't have negative length)
Triangle 2: AB = 1, BC = 1