find the lateral surface area of a right triangular prism if its altitude is five inches and its base is an isoceles right triangle with hypotenuse of length 10 inches
To find the lateral surface area of a right triangular prism, you need to determine the area of all the lateral faces. In this case, the prism has three rectangular lateral faces.
First, let's determine the dimensions of one of the rectangular faces. Since the base is an isosceles right triangle with a hypotenuse of 10 inches, that means the two legs of the triangle are equal.
Using the Pythagorean theorem, we can find the length of each leg:
\( a^2 + b^2 = c^2 \) (where a and b are the legs, and c is the hypotenuse)
\( a^2 + a^2 = 10^2 \)
\( 2a^2 = 100 \)
\( a^2 = 50 \)
\( a = \sqrt{50} \)
\( a = 5 \sqrt{2} \)
Since the triangular base is an isosceles triangle, the other side of the right triangle is also \( 5 \sqrt{2} \) inches.
Now, to find the lateral surface area, you need to calculate the area of one lateral face and multiply it by the number of such faces (which is 3 in this case).
The area of a rectangle is given by the formula \( A = lw \), where \( l \) represents the length and \( w \) represents the width.
Since the length of each rectangular face is equal to the altitude of the prism (5 inches), and the width is equal to one side of the right triangle (\( 5 \sqrt{2} \) inches), the area of one rectangular face is:
\( A = 5 \times 5 \sqrt{2} \)
\( A = 25 \sqrt{2} \) square inches
Since there are three such faces, the total lateral surface area of the right triangular prism is:
\( Total \, Lateral \, Surface \, Area = 3 \times 25 \sqrt{2} \)
\( Total \, Lateral \, Surface \, Area = 75 \sqrt{2} \) square inches