Triangles C and D are similar. The area of triangle C is 47.6 in(2). The base of Trianfle D is 6.72 in. Each Dimension of D is 6/5 the corresponding dimension of C. What is the Height of D?
The area of D will be (6/5)² times the area of C, which is (6/5)² x 47.6"² = 68.544"² (don't forget that you need to square that factor of (6/5), because it applies to both the base AND the height). The base of D is 6.72", and the area of a triangle is half the base times the height. You've now got both the area and the base... and the height is half the area divided by the base.
Er... if the area is half the base times the height, then the height must be TWICE the area divided by the base. Sorry!
thank you soooo much!
To find the height of triangle D, we need to use the fact that the area of a triangle is equal to one-half the base multiplied by the height.
First, let's find the base and height of triangle C:
Given that the area of triangle C is 47.6 in^2, we can set up the equation:
47.6 = (1/2) x base of C x height of C.
Since we don't know the base of C, let's solve for it:
47.6 = (1/2) x base of C x height of C
Multiplying both sides of the equation by 2:
95.2 = base of C x height of C
Next, let's find the base and height of triangle D:
Given that each dimension of triangle D is 6/5 the corresponding dimension of triangle C, we can find the base and height of D using the following equations:
base of D = (6/5) x base of C
height of D = (6/5) x height of C
Now, substitute the values we know:
base of D = (6/5) x base of C = (6/5) x [base of D]
Notice that the base of D is both unknown and included in this equation. This is an example of a dependent variable, so we need to solve for the base of D first.
To do this, divide both sides of the equation by (6/5):
base of D = (6/5) x [base of D] / (6/5)
Simplifying the expression:
base of D = [base of D]
Now, we realize that the base of D can be any value, since both sides of the equation are equal. However, we do know that the base of D is given as 6.72 in. Therefore, we conclude that the base of D is 6.72 in.
Now we can find the height of D using the equation:
height of D = (6/5) x height of C
Substituting the known values:
height of D = (6/5) x [height of D]
Again, we have the height of D on both sides of the equation, so let's solve for it:
height of D = (6/5) x [height of D] / (6/5)
height of D = [height of D]
Similar to the previous step, we realize that the height of D can be any value. However, we are given that the height of C is a value that corresponds to an area of 47.6 in^2. Therefore, we want to find the corresponding height of D that will give us the same ratio of areas.
Knowing that the ratio of corresponding sides is 6/5, and the ratio of corresponding areas is the square of the ratio of sides, we can set up the equation:
[(6/5) x height of D]^2 = 47.6
Simplifying the expression:
(36/25) x height of D^2 = 47.6
Dividing both sides of the equation by (36/25):
height of D^2 = 47.6 / (36/25)
height of D^2 = 66.11
Taking the square root of both sides of the equation:
height of D = √66.11
Calculating this square root, the height of D is approximately 8.12 inches.