Applying the properties of circumcenter of a triangle and the Pythagorean Theorem, the measure of each given segments are calculated as:
8. TN = 22
9. RT = 12.8
Recall:
- The point where the three perpendicular bisectors of a triangle meet is known as the circumcenter of the triangle.
- The perpendicular bisectors form right angles with each side it bisects. This means that there are two opposite right angles formed at point Q, S, and R.
- The circumcenter is also the point that is equidistant to all three vertices of the triangle. This means that, in the diagram given, MT, TN, and TP are equal.
- Pythagorean theorem can be used to solve for the missing side of a right triangle. It states that, if c is the hypotenuse (longest side that is directly opposite to the right triangle), and a and b are the two other smaller sides, therefore: [tex]c^2 = a^2 + b^2[/tex]
8. Find TN
Given,
Thus,
6x - 56 = 3x - 17 (MT = TN = TP, because they are equidistant from the circumcenter, T.)
6x - 3x = 56 - 17
3x = 39
x = 13
TN = TP = 3x - 17
TN = 3x - 17
TN = 3(13) - 17
TN = 22
9. Find RT
Given,
- RN = 14
- TN = 4x - 17
- TP = x + 10
Thus,
4x - 17 = x + 10 (TN = TP)
4x - x = 17 + 10
3x = 27
x = 9
TN = 4x - 17
TN = 4(9) - 17
TN = 19
To find RT, apply the Pythagorean Theorem
[tex]RT = \sqrt{TN^2 - RN^2}[/tex]
[tex]RT = \sqrt{19^2 - 14^2}\\\\\mathbf{RT = 12.8}[/tex]
In summary, applying the properties of circumcenter of a triangle and the Pythagorean Theorem, the measure of each given segments are calculated as:
8. TN = 22
9. RT = 12.8
Learn more about circumcenter of a triangle here:
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