Question - Solving Geometry Problem with Similar Triangles
Solution:
This geometry problem can be solved using the properties of similar triangles.From the image, there are two triangles: a larger right-angled triangle, and a smaller one inside it. Both are right-angled. Given that they share an acute angle, we can conclude that they are similar by the AA (Angle-Angle) criterion.When two triangles are similar, the ratios of their corresponding sides are proportional. The corresponding sides of the smaller triangle (with hypotenuse 3 and an unknown adjacent side to the right angle x) and the larger triangle (with hypotenuse 5 and an adjacent side to the right angle 3) should be in proportion.So we set up our proportion using the hypotenuses and one pair of corresponding sides:For the larger triangle, the sides are 5 (hypotenuse) and 3 (adjacent side). For the smaller triangle, they are 3 (hypotenuse) and x (adjacent side). Setting up a proportion, we get:5 / 3 = 3 / x => 5x = 9 (Cross multiplying)=> x = 9 / 5=> x = 1.8 or 1 4/5Therefore, x is 1.8 or 1 4/5 in decimal and fractional forms, respectively. Since the question asked for the simplest radical form and both 1.8 and 1 4/5 are neither irrational nor do they have square roots, we can simply state that x = 1.8 or x = 1 4/5 as our final answer.
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