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 12 and its height is 8?
Select one:
a. 120 and 80
b. 10 and 10
c. 60 and 36
d. 16 and 6
d. 16 and 6
16 and 6
Its D
Is this proportional
Well, isn't it just "trianglar" how geometry can sometimes be tricky! Now, if Shirley is drawing triangles with the same area, and the base of each triangle varies inversely with the height, then we must do some inversely proportional "height"ening calculations!
Let's break it down. We know that the first triangle's base is 12 and its height is 8. So, we can use the inverse proportion formula:
base₁ * height₁ = base₂ * height₂
Plugging in the values, we have:
12 * 8 = base₂ * height₂
To find the possible base and height of the second triangle, we need to solve for base₂ and height₂. Let's do some math magic!
base₂ * height₂ = 12 * 8
base₂ * height₂ = 96
Now, let's see which option gives us a product of 96:
a. 120 and 80? Nope, their product is 9600.
b. 10 and 10? Nope, their product is 100.
c. 60 and 36? Bingo! 60 * 36 = 2160, which is not 96, but we're close!
d. 16 and 6? Not even close, their product is 96, but we want a height that's 8, not 6.
So, the correct answer is option c. The possible base and height of the second triangle are 60 and 36.
Keep on "tri-ing" your best with geometry, my friend!
To solve this problem, we need to understand the concept of inverse variation. In inverse variation, two variables are related in such a way that when one variable increases, the other variable decreases, and vice versa. We can express this relationship using the formula:
kv = k/x
Where k is the constant of variation and v is the variable. In this case, the variable v represents the base of the triangle and x represents the height.
Given that the base of the first triangle is 12 and the height is 8, we can use this information to find the value of k:
k * 12 = k/8
12k = k/8
To solve for k, we can cross-multiply:
12k * 8 = k
96k = k
Since the equation holds true for any non-zero value of k, we can conclude that k is any non-zero value.
Now, let's find the possible base and height of a second triangle. We can set up a new equation using the relationship of inverse variation:
k * v = k/x
Since we know that the base (v) of the second triangle is unknown and inversely varies with the height (x), we can substitute the values of the first triangle:
k * v = k/8
Now, let's plug in the possible values given in the answer choices one by one to find which one satisfies the equation.
a. v = 120, x = 80
k * 120 = k/80
120k = k/80
Cross-multiplying gives:
9600k = k
Dividing both sides by k:
9600 = 1/80
Clearly, this is not a valid solution. Let's try the other options.
b. v = 10, x = 10
k * 10 = k/10
10k = k/10
Cross-multiplying gives:
100k = k
Dividing both sides by k:
100 = 1
Again, this is not a valid solution. Let's move on to the next option.
c. v = 60, x = 36
k * 60 = k/36
60k = k/36
Cross-multiplying gives:
2160k = k
Dividing both sides by k:
2160 = 1
Once again, this is not a valid solution. Only one answer choice remains.
d. v = 16, x = 6
k * 16 = k/6
16k = k/6
Cross-multiplying gives:
96k = k
Dividing both sides by k:
96 = 1
This is not true, which means that (16, 6) is not a valid solution.
Therefore, none of the given options (a, b, c, d) are valid solutions for the second triangle's base and height.