Two similar triangles have a pair of corresponding sides of length 12 meters and 8 meters. The larger triangle has a perimeter of 48 meters and an area of 180 square meters. Find the perimeter P and area A of the smaller triangle.
the side length of the smaller triangle is 2/3 that of the larger triangle
so the perimeter of the smaller triangle is 2/3 the perimeter of the larger triangle
and the area of the smaller triangle is (2/3)^2 the area of the larger triangle
To find the perimeter P and area A of the smaller triangle, we can use the concept of similarity between triangles.
First, let's denote the length of the corresponding side of the smaller triangle as x meters.
Since the two triangles are similar, we can set up a proportion using the corresponding sides:
(Length of corresponding side of larger triangle) / (Length of corresponding side of smaller triangle) = (Perimeter of larger triangle) / (Perimeter of smaller triangle)
12 / x = 48 / P
Cross-multiplying the above equation, we get:
12P = 48x
Dividing both sides by 12, we obtain:
P = 4x
Therefore, the perimeter of the smaller triangle is 4 times the length of the corresponding side.
Next, let's set up a proportion for the areas of the two triangles:
(Area of larger triangle) / (Area of smaller triangle) = (Length of corresponding side of larger triangle)^2 / (Length of corresponding side of smaller triangle)^2
180 / A = 12^2 / x^2
Simplifying the equation, we have:
180x^2 = 12^2 * A
180x^2 = 144A
Dividing both sides by 180, we get:
x^2 = (144A) / 180
Simplifying further, we obtain:
x^2 = 4A / 5
Taking the square root of both sides, we get:
x = √(4A / 5)
Therefore, the length of the corresponding side of the smaller triangle is the square root of 4A divided by 5.
In summary, the perimeter P of the smaller triangle is 4 times the length of the corresponding side, and the area A of the smaller triangle is given by A = 5x^2 / 4, where x is the length of the corresponding side.
To solve this problem, we can use the concept of similarity of triangles.
Two triangles are similar if their corresponding angles are equal and the corresponding sides are in proportion. In this case, we are given that the triangles are similar, and we also know the length of one pair of corresponding sides.
Let's denote the length of the other pair of corresponding sides in the smaller triangle as x and y meters, respectively. Since the triangles are similar, we can set up a proportion:
12/8 = x/12
Cross-multiplying this proportion, we get:
12x = 12*8
12x = 96
x = 8 meters
Now, we can find the ratio of the perimeters of the two triangles:
Perimeter of the larger triangle / Perimeter of the smaller triangle = Corresponding side lengths of the larger triangle / Corresponding side lengths of the smaller triangle
48/P = 12/8
Cross-multiplying this proportion, we get:
48*8 = 12P
384 = 12P
P = 384/12
P = 32 meters
So, the perimeter of the smaller triangle is 32 meters.
To find the area of the smaller triangle, we know that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding side lengths:
Area of the larger triangle / Area of the smaller triangle = (Corresponding side lengths of the larger triangle / Corresponding side lengths of the smaller triangle)^2
180/A = (12/8)^2
Simplifying, we get:
180/A = 9/4
Cross-multiplying this proportion, we get:
9A = 180*4
9A = 720
A = 720/9
A = 80 square meters
So, the area of the smaller triangle is 80 square meters.
In summary:
Perimeter of the smaller triangle (P) = 32 meters
Area of the smaller triangle (A) = 80 square meters
Since S2/S1 = 8/12 = 2/3,
P2/P1 = 2/3
A2/A1 = (2/3)^2