a. To find X, you will need to add the arc measures together and set this expression equal to the total degrees of a circle and then solve for X. Must show all work.

b. Here are the steps to take (show all work)

1. Plug your x-value from a into each of the arc measures to find what each ars measures.
2. Apply the Inscribed Angles Theorem to then find <P, <R, and <S
3 Use the angle measure from step 2 above to determine what type of triangle yoy have.

the picture is a circle with triangle inscribed top point being P the bottom left being Q and bottom right being R. The left side (8x-10), right side (6x), and bottom (10x+10).
I have been given the answer but I need to know how to get to that.Can someone PLEASE help!!
Answer:
a. x=15
b. scalene the arc measures are 110, 90, 160, the arcs are not congruent neither are the chords that intercept.

At first, I was having difficulty understanding the given. I thought the PQ, PR and QR are the lengths of the triangle, but they are actually the arc (angle measure).

Anyway, to get x, we know that the measure of a circle is 360 degrees. We get the sum of arcs PQ, PR and QR and equate it to 360:
(8x - 10) + 6x + (10x + 10) = 360
24x = 360
x = 15
now we substitute this back to the measure of the arcs:
PQ = 8*15 - 10 = 110 degrees
PR = 6*15 = 90 degrees
QR = 10*15 + 10 = 160 degrees

Then, recall inscribed angle theorem. The measure of an inscribed angle (in this case, an interior angle of the triangle) is half the measure of the arc it subtends. Thus the interior angles of the triangles are half of PQ, PR and QR, which are 55, 45 and 80.
The triangle is SCALENE because there are no angles of equal measure.

hope this helps~ :)

Thank you didn't think I was ever gonna get help with this one. You explained it so well that it makes sense now.Thank you again

a. To find X, you will need to add the arc measures together and set this expression equal to the total degrees of a circle and then solve for X. Must show all work.

Let's start by adding the arc measures together:

Arc measure PQ -> 8x - 10
Arc measure QR -> 6x
Arc measure RP -> 10x + 10

Total arc measures:
(8x - 10) + (6x) + (10x + 10)

Now, we set the sum of the arc measures equal to 360 degrees (since a circle has 360 degrees):

(8x - 10) + (6x) + (10x + 10) = 360

Combine like terms:
8x + 6x + 10x - 10 + 10 = 360

Combine like terms again:
24x = 360

Divide both sides by 24:
x = 15

Therefore, X is 15.

b. Here are the steps to take (show all work):

1. Plug your x-value from part a into each of the arc measures to find what each arc measures:
Arc measure PQ = 8(15) - 10 = 110 degrees
Arc measure QR = 6(15) = 90 degrees
Arc measure RP = 10(15) + 10 = 160 degrees

2. Apply the Inscribed Angles Theorem to then find <P, <R, and <S:
The Inscribed Angles Theorem states that an angle formed by two chords in a circle is equal to half the sum of the intercepted arcs.

Angle P = 0.5 * Arc measure RP = 0.5 * 160 = 80 degrees
Angle R = 0.5 * Arc measure PQ = 0.5 * 110 = 55 degrees
Angle S = 0.5 * Arc measure QR = 0.5 * 90 = 45 degrees

3. Use the angle measure from step 2 above to determine what type of triangle you have:
Since the angles P, R, and S are all different, the triangle formed by the chords is scalene. Additionally, the arc measures are 110, 90, and 160 degrees, which means the arcs are not congruent. Therefore, both the arcs and the chords that intercept them are not congruent.

To find the value of x, we will use the given information about the arc measures of the circle.

a. Here are the steps to find the value of x:

1. Add the arc measures together:
(8x - 10) + (6x) + (10x + 10) = 360
Simplify the equation by combining like terms:
24x = 360

2. Set the expression equal to the total degrees of a circle:
24x = 360

3. Solve for x:
Divide both sides of the equation by 24:
x = 15

Therefore, the value of x is 15.

b. To find the measures of <P, <Q, and <R, we can substitute the value of x into the given arc measures.

1. Substitute x = 15 into the arc measures:
<P = 8x - 10 = 8(15) - 10 = 110
<Q = 6x = 6(15) = 90
<R = 10x + 10 = 10(15) + 10 = 160

2. Apply the Inscribed Angles Theorem:
The theorem states that an inscribed angle is equal in measure to half of the intercepted arc.

3. Find the type of triangle:
Since the measures of <P, <Q, and <R are 110, 90, and 160 respectively, and the arcs are not congruent, we can determine that the triangle formed by the points P, Q, and R is a scalene triangle. This means that none of the sides or angles are congruent.

So, the final answer is:
a. x = 15
b. The triangle formed by the points P, Q, and R is a scalene triangle, with angle measures of 110, 90, and 160 degrees, respectively.