The area of triangle B is 64 times greater than the area of triangle A.The perimeter of triangle B is how many times greater than the perimeter of triangle A?
how many times greater than the perimeter of triangle A?
A. 8 times greater
B.64 times greater
C.128 times greater
D.256 times greater
area=sqrt(s(s-a)(s-b)(s-c)) where s is the semi perimeter.
so area=f(sqrt s^4) or f(s^2)
so if area increases by 64, s must have increased by 8
128
To find the answer, let's consider the relationships between the areas and perimeters of triangles.
The area of a triangle is given by the formula:
Area = (1/2) * base * height.
Let's say the base and height of triangle A are bA and hA, respectively. Therefore, the area of triangle A is:
Area A = (1/2) * bA * hA.
According to the question, the area of triangle B is 64 times greater than triangle A. Thus, the area of triangle B is:
Area B = 64 * Area A
= 64 * 0.5 * bA * hA
= 32 * bA * hA.
Now, let's consider the relationship between the perimeters of triangles. The perimeter of a triangle is given by the formula:
Perimeter = side1 + side2 + side3.
Let's say the sides of triangle A are sA1, sA2, and sA3. Therefore, the perimeter of triangle A is:
Perimeter A = sA1 + sA2 + sA3.
According to the question, we need to find the factor by which the perimeter of triangle B is greater than triangle A.
Since the areas of the triangles are proportional, we can assume that the bases and heights of triangle B are 8 times larger than triangle A, making the sides also 8 times larger. Therefore, the sides of triangle B are:
sB1 = 8 * sA1, sB2 = 8 * sA2, sB3 = 8 * sA3.
Now, we can calculate the perimeter of triangle B:
Perimeter B = sB1 + sB2 + sB3
= (8 * sA1) + (8 * sA2) + (8 * sA3)
= 8 * (sA1 + sA2 + sA3)
= 8 * Perimeter A.
Since the perimeter of triangle B is 8 times greater than the perimeter of triangle A, the correct answer is A. 8 times greater.
To find the answer, we need to understand the relationship between the areas and perimeters of triangles.
Let's denote the area of triangle A as A_A and the area of triangle B as A_B. Similarly, let's denote the perimeter of triangle A as P_A and the perimeter of triangle B as P_B.
Given that the area of triangle B is 64 times greater than the area of triangle A, we can express this mathematically as:
A_B = 64 * A_A
Now, we want to find the relationship between the perimeters. Let's consider the following property:
- If the side lengths of a triangle are multiplied by a certain factor, the perimeter of the triangle will also be multiplied by the same factor.
This means that if we multiply the side lengths of triangle A by a factor of x, the perimeter of triangle A (P_A) will also be multiplied by x.
Now, let's consider the relationship between the side lengths of triangle A and triangle B. Since the area of triangle B is 64 times greater than the area of triangle A, we can infer that the side lengths of triangle B are √64 = 8 times greater than the side lengths of triangle A.
Therefore, the perimeter of triangle B (P_B) will be 8 times greater than the perimeter of triangle A (P_A).
So, the correct answer is:
A. 8 times greater.