Question

The base of a tetrahedron is a triangle whose sides have lengths 10, 24, and 26. The altitude of the tetrahedron is 20. Find the area of a cross section whose distance from the base is 15.

Answer 1

The side lengths of the base make it a right triangle, with legs 10 and 24. Then, the area of that base is (1/2)(10)(24) = 120.

Now, the cross-section is 15/20 = 3/4 of the way from the base to the peak. The cross-section is a right triangle similar to the base; its side lengths are 1/4 of the corresponding side lengths of the base.

So the legs of the cross-section are 5/2 and 6; the area is (1/2)(5/2)(6) = 15/2.

Notice the area of the cross-section can be easily determined without finding the side lengths of the sides. The ratio of similarity between the cross-section and the base is 1:4, so the ratio of their areas is 1:16. So the area of the cross-section is (1/16)*120 = 120/16 = 15/2.

So that, the correct answer is

`(15)/(2)=7.5`