In the given triangular prism, AB= 3 cm ,DF= 4cm, BC= 5 cm and CF=12cm ,then find the area of rectangle surface of the prism
To find the area of the rectangle surface of the prism, we need to determine the length and width of the rectangle.
In a triangular prism, the rectangular surface is formed by the two parallel triangular bases. Let's call the vertices of one triangular base A, B, and C, and the vertices of the other triangular base D, E, and F.
From the given information, we know that AB = 3 cm, DF = 4 cm, BC = 5 cm, and CF = 12 cm.
To find the length of the rectangle, we need to find the distance between A and D, which is the altitude of the triangular base ADF. We can use the Pythagorean theorem to calculate this distance.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Since AD and AF are the two legs of a right triangle:
AD^2 + DF^2 = AF^2
Substituting the known values:
AD^2 + 4^2 = 12^2
AD^2 + 16 = 144
AD^2 = 144 - 16
AD^2 = 128
AD = √128
AD = 8√2 cm
So, the length of the rectangle is 8√2 cm.
To find the width of the rectangle, we need to find the distance between A and C. This is simply the length AB, which is given as 3 cm.
Therefore, the width of the rectangle is 3 cm.
Now that we know the length (8√2 cm) and width (3 cm) of the rectangle, we can calculate its area:
Area = Length x Width
Area = 8√2 cm x 3 cm
Area = 24√2 cm^2
Thus, the area of the rectangle surface of the given triangular prism is 24√2 cm^2.