Question

a package is in the shape of a triangular prism. the bases are right triangles with perpendicular legs measuring 9 centimeters and 12 centimeters. the distance between the bases is 10 centimeters. what is the surface area of the triangular prism? 210 square centimeters 318 square centimeters 360 square centimeters 468 square centimeters

Explanation:

Step1: Find hypotenuse of base triangle

Use Pythagorean theorem:
$$c = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \text{ cm}$$

Step2: Calculate area of 2 triangular bases

Area of one right triangle: $\frac{1}{2} \times 9 \times 12 = 54$ cm². For two bases:
$$2 \times 54 = 108 \text{ cm}^2$$

Step3: Calculate lateral surface area

Sum areas of 3 rectangular faces:
$$(9 \times 10) + (12 \times 10) + (15 \times 10) = 90 + 120 + 150 = 360 \text{ cm}^2$$

Step4: Total surface area

Add base area and lateral area:
$$108 + 360 = 468 \text{ cm}^2$$

Answer:

468 square centimeters