Answer:
10. x = 4
11. UT = 44
Explanation:
You want the value of x and the length of segment UT that bisects chord PR at point U into halves marked UP = (9x -3) and UR = (7x +5). The radius of circle T is 55 units.
10. Chord
Point U is the midpoint of chord PR, so ...
UP = UR
9x -3 = 7x +5
2x = 8 . . . . . . . . . add 3-7x
x = 4 . . . . . . . . divide by 2
11. UT
Segments UP, UT and TP form a right triangle with one leg ...
UP = 9(4) -3 = 33
and hypotenuse 55.
We observe that the ratio of these measures being 33/55 indicates triangle PUT is a 3-4-5 right triangle, with scale factor 11. That means UT is 4·11 = 44 units.
UT = 44 units
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Additional comment
Recognizing triangles made from Pythagorean triples, such as {3, 4,5}, {5, 12, 13}, {7, 24, 25}, or {8, 15, 17} can save a lot of work. If you don't recognize these, you can find the missing side length from ...
TP² = UP² +UT²
UT = √(TP² -UP²) = √(55² -33²) = √1936 = 44
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