Question 195700: There are two similar right triangles.
The larger triangles hypotenuse is 15 and a leg is 3x, and the other is 3x+3. The smaller triangle has a hypotenuse of 5 one leg is x and the other is x+1.
How would you write an equation expressing the relationship among the length of the sides of the triangles for the smaller and the larger triangle?
Answer by J2R2R(94)   (Show Source):
You can put this solution on YOUR website! There are two similar right angled triangles.
The larger triangles hypotenuse is 15 and a leg is 3x, and the other is 3x+3. The smaller triangle has a hypotenuse of 5 one leg is x and the other is x+1.
How would you write an equation expressing the relationship among the length of the sides of the triangles for the smaller and the larger triangle?
Going by Pythagoras’ Theorem, the square on the hypotenuse is equal to the sum of the squares of the other two sides, which means:
For the larger triangle, 15^2 = (3x)^2 + (3x+3)^2
225 = 9x^2 + 9x^2 + 18x + 9 = 18x^2 + 18x + 9 = 9 (2x^2 + 2x + 1)
25 = 2x^2 + 2x + 1 (cancelling by 9)
giving 0 = 2x^2 + 2x – 24
and then 0 = x^2 + x – 12 (cancelling by 2)
We are left with x^2 + x – 12 = 0 = (x + 4)(x – 3) giving x = 3 or -4 but we cannot have a negative length, so the solution is x = 3 units whatever the units are.
This is the same solution for the smaller triangle since they are similar triangles and every factor has been scaled down by a factor of 3 squared (9) as 3 is the ratio of the larger triangle to the smaller triangle. We would have arrived at 25 = 2x^2 + 2x + 1 without cancelling by 9 if we did to the small triangle what we did to the large triangle.
So the sides of the triangles are:
Large: 15, 3x and 3x + 3 giving 15, 9 and 12
Small: 5, x and x + 1 giving 5, 3 and 4.
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