Look at the given triangles. There are two right triangles. One red and one blue. The blue triangle has the following dimensions: To create the right angle the vertical side has the length of 4x + 2 and the width of 5x – 4. The remaining segment is 7x + 7. The red triangle is also a right triangle. To create the right angle the vertical segment measures x + 3 with its horizontal width of x + 7. The remaining segment measures 2x – 5
a. Write an expression in simplest form for the perimeter of each triangle.
b. Write another expression in simplest form that shows the difference between the perimeter of the larger triangle and the perimeter of the smaller triangle.
c. Find the perimeter for each triangle when x = 3
how would I write this
a. Perimeter of blue triangle = (4x + 2) + (5x – 4) + (7x + 7) = 16x + 5
Perimeter of red triangle = (x + 3) + (x + 7) + (2x – 5) = 4x + 5
b. Difference between perimeters = (16x + 5) - (4x + 5) = 12x
c. When x = 3,
Perimeter of blue triangle = (4(3) + 2) + (5(3) – 4) + (7(3) + 7) = 52
Perimeter of red triangle = (3 + 3) + (3 + 7) + (2(3) – 5) = 10
Therefore, the perimeter of the blue triangle is 52 and the perimeter of the red triangle is 10 when x=3.
For the blue triangle, it can only be a right triangle if
(4x + 2)^2 + (5x – 4)^2 = (7x + 7)^2
But this is only true if x = (-61±√3489)/8
and both those values give a negative value for 5x-4
So asking for the perimeter when x=3 is pointless, since that means the original premise is false.
I didn't bother to check the red triangle.
Thank you for catching that mistake. You are correct, the dimensions given for the blue triangle do not form a valid right triangle since it violates the Pythagorean theorem. I apologize for the error in my earlier response.
a. To find the perimeter of a triangle, we need to add up the lengths of all its sides.
For the blue triangle:
Perimeter = vertical side length + width + remaining segment
Perimeter = (4x + 2) + (5x - 4) + (7x + 7)
Perimeter = 4x + 2 + 5x - 4 + 7x + 7
Perimeter = 16x + 5
For the red triangle:
Perimeter = vertical segment + horizontal width + remaining segment
Perimeter = (x + 3) + (x + 7) + (2x - 5)
Perimeter = x + 3 + x + 7 + 2x - 5
Perimeter = 4x + 5
b. To find the difference between the perimeters of the larger triangle (blue) and the smaller triangle (red), we subtract the perimeter of the smaller triangle from the perimeter of the larger triangle.
Difference = Perimeter of blue triangle - Perimeter of red triangle
Difference = (16x + 5) - (4x + 5)
Difference = 16x - 4x
Difference = 12x
c. To find the perimeter for each triangle when x = 3, we substitute x = 3 into the expressions we derived in part a.
For the blue triangle:
Perimeter = 16x + 5
Perimeter = 16(3) + 5
Perimeter = 48 + 5
Perimeter = 53
For the red triangle:
Perimeter = 4x + 5
Perimeter = 4(3) + 5
Perimeter = 12 + 5
Perimeter = 17