+274 1. A gecko is in a room that is 12 feet long, 10 feet wide and 8 feet tall. The gecko is currently on a side wall (\(10^{\prime}\) by \(8^{\prime}\)), one foot from the ceiling and one foot from the back wall (\(12^{\prime}\) by \(8^{\prime}\)). The gecko spots a fly on the opposite side wall, one foot from the floor and one foot from the front wall. What is the length of the shortest path the gecko can take to reach the fly assuming that it does not jump and can only walk across the ceiling and the walls? Express your answer in simplest radical form.
2. Five points A, B, C, D, and O lie on a flat field. A is directly north of O, B is directly west of O, C is directly south of O, and D is directly east of O. The distance between C and D is 140 m. A hot-air balloon is positioned in the air at H directly above O. The balloon is held in place by four ropes HA, HB, HC, and HD. Rope has length 150 m and rope HC has length 130 m. How high is the balloon above the field (that is, the length of OH)? https://latex.artofproblemsolving.com/7/4/3/7438def2a4f1ca09215d45afe33e60a68d704b74.png
1. The shortest path the gecko can take is a diagonal line across the room. The length of this diagonal line is the hypotenuse of a right triangle with legs of length 10 and 12. The length of the hypotenuse is given by the Pythagorean Theorem:
a^2 + b^2 = c^2
where a and b are the lengths of the legs and c is the length of the hypotenuse. Substituting in the lengths of the legs, we get:
10^2 + 12^2 = c^2
100 + 144 = c^2
244 = c^2
c = \sqrt{244} = 2\sqrt{61}
Therefore, the length of the shortest path the gecko can take is 2*sqrt(61) feet.
