Respuesta :

We have been given that SU is diameter of our circle so measure of arc ST will be equal measure of central angle SOT.

Diameter is longest line segment of a circle passing through center of circle. So angles formed by our diameter (straight line) will add up-to 180 degrees.

Let us set sum of measures of our angles equal to 180. We have been given that angle SOR and UOT are congruent angles, so we will set our equation as:

[tex]2(9x+5)+13x+15=180[/tex]

Upon distributing 2 we will get,

[tex]18x+10+13x+15=180[/tex]

[tex]31x+25=180[/tex]

[tex]31x=180-25[/tex]

[tex]31x=155[/tex]

[tex]x=\frac{155}{31} =5[/tex]

We can find measure of angle SOT by sum of measures of angle ROT and UOT.

[tex]\angle SOT=(13x+15)+(9x+5)[/tex]

[tex]\angle SOT=13x+15+9x+5[/tex]

Let us combine like terms.

[tex]\angle SOT=22x+20[/tex]

Let us substitute x=5 in our given equation.

[tex]\angle SOT=22\cdot 5+20[/tex]

[tex]\angle SOT=110+20=130[/tex]

Therefore, measure of arc ST will be 130 degrees.

akposevictor

The measure of arc ST, based on the central angle and intercepted arc relationship is: 130°.

Relationship between Central Angle and the Intercepted Arc

  • The measure of a central angle = measure of the intercepted arc.
  • A semicircle = 180°.

Thus:

m∠TOU = m∠SOR (congruent angles given)

2(m∠SOR) + m∠ROT = 180°

  • Substitute

2(9x + 5) + (13x + 15) = 180

18x + 10 + 13x + 15 = 180

31x + 25 = 180

31x = 180 - 25

31x = 155

x = 5

measure of arc ST = m∠ROT + m∠SOR

  • Substitute

measure of arc ST = 13x + 15 + 9x + 5

  • Plug in the value of x

measure of arc ST = 13(5) + 15 + 9(5) + 5

measure of arc ST = 130°

Learn more about central angle and intercepted arc relationship on:

https://brainly.com/question/25889947

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