Answer:
KS = 25
KP = 17
Step-by-step explanation:
In ratio units, the midsegment is (3+7)/2 = 5, so each ratio unit stands for 4 units by which the trapezoid dimensions are measured. Thus PS = 12 and KT = 28.
Then the segment KP is the hypotenuse of a right triangle with legs 8 and 15, so is 17 units. The segment KS is the hypotenuse of a right triangle with legs (8+12) and 15, so is 25 units.
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It is helpful, but not essential, to recognize that the Pythagorean triples (8, 15, 17) and (3, 4, 5) come into play in this problem.
The trapezoid is isosceles. If you cut the middle 12 units out of the trapezoid's bases, you have an isosceles triangle with a height of 15 units and a base of 16 units. That triangle can be divided by its altitude into two right triangles, each with legs 8 and 15 units. The hypotenuse (KP) is then √(8²+15²) = √289 = 17.
The diagonal is the hypotenuse of a right triangle still with height 15, but the other leg is 12 units longer than the 8 used in the above calculation. You can recognized legs of 15 and 20 as being 5 times those of a (3, 4, 5) right triangle, so the diagonal length (KS) will be 5·5 = 25 units. Or, you can calculate it using the Pythagorean theorem: √(15²+20²) = √625 = 25.
