Shirley is drawing triangles that have the same area. The base of each triangle varies inversely with the height. What are the possible base and height of a second triangle if the first triangle's base is 10 and its height is 6
any pair of numbers whose product is 60
To solve this problem, we can use the concept of inverse variation.
Inverse variation means that as one variable increases, the other variable decreases, and vice versa, while their product remains constant.
Let's denote the base of the second triangle as x and its height as y.
Given that the base of the first triangle is 10 and its height is 6, we can set up the following proportion:
10 * 6 = x * y
Now, if we solve this proportion for x, we get:
x = (10 * 6) / y
Since the area of the triangle remains the same, the product of the base and height for both triangles should be equal. So, the second triangle can have any base and height combination that satisfies this equation.
For example, if we choose a height of 12 for the second triangle, the equation becomes:
x = (10 * 6) / 12
x = 5
Therefore, one possible base and height for the second triangle could be a base of 5 and a height of 12.
To find the possible base and height of a second triangle that has the same area as the first triangle, we need to use the concept of inverse variation.
In this case, let's denote the base of the second triangle as 'b2' and its height as 'h2'. We know that the area of a triangle is given by the formula: Area = (1/2) × base × height.
Since the triangles have the same area, we can set up the following equation using the given values for the first triangle:
(1/2) × 10 × 6 = (1/2) × b2 × h2
Simplifying the equation, we have:
30 = (1/2) × b2 × h2
To solve for the possible values of b2 and h2, we need to consider the inverse variation relationship between the base and height. Inverse variation tells us that as one value increases, the other value decreases, while their product remains constant.
So, if the base of the first triangle is 10 and its height is 6, we can set up a proportion to solve for the second triangle:
10 × 6 = b2 × h2
60 = b2 × h2
Therefore, the possible base and height of the second triangle could be any two values that multiply to 60. For example, the second triangle could have a base of 5 and a height of 12, or a base of 2 and a height of 30, among other combinations.