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?
We know that the area of a triangle is given by the formula A = 1/2 × base × height. Since the area of both triangles is the same, we can write:
1/2 × b1 × h1 = 1/2 × b2 × h2
where b1 and h1 are the base and height of the first triangle, and b2 and h2 are the base and height of the second triangle.
We also know that the base and height of the second triangle vary inversely with each other, so we can write:
b2 = k/h2
where k is a constant of proportionality. Substituting this into the equation above, we get:
1/2 × b1 × h1 = 1/2 × (k/h2) × h2
Simplifying, we get:
b1 × h1 = k
Substituting the values of b1 and h1 from the first triangle, we get:
10 × 6 = k
k = 60
Therefore, the second triangle's base and height are related by the equation:
b2 = 60/h2
We can choose any value of h2 and calculate the corresponding value of b2. For example:
- If h2 = 4, then b2 = 60/4 = 15
- If h2 = 5, then b2 = 60/5 = 12
- If h2 = 8, then b2 = 60/8 = 7.5
So the possible base and height of the second triangle are:
- Base = 15, Height = 4
- Base = 12, Height = 5
- Base = 7.5, Height = 8
(Note that the units of measurement are not specified in the problem, so we can't determine them from the given information.)