--------------- QUESTION FOLLOWS --------------------------
Can someone Please help I definitely do not know how to do this.
A)Find x, to find x you need to add the arc measures together and set the expression equal to the total degrees of circle then solve for X, must show all steps.
B)Is the triangle equilateral, isosceles, or scalene, explain, plug your x-value from A into each of the arc measures to find what each arc measures.
Apply the Inscribed Angles Theorem to find <P, <R, <S.
Use the angle measure from step 2 above to determine what type of triangle you have.
The picture is a circle with a triangle inside, QP=(8x-10) and PR=(6x), and QR=(10x+10) QP and PR are the sides and QR is the bottom.
Sure hope someone can help.
According to the wording of part a) your description of
QP= 8x-10, PR = 6x etc
must have said
arc QP = 8x-10 etc
so 8x-10 + 6x + 10x+10 = 360
24x = 360
x = 15
so arc QP = 8(15) - 10 = 110°
PR = 90°
QR = 160°
By the inscribed angle theorem, the angle opposite the arc must be half the central angle subtended by the arc. Then....
angle R = 55°
angle Q =45°
angle P = 80°
looks like plain old scalene triangle
To find x in this problem, you need to add the arc measures together and set the expression equal to the total degrees of a circle. Then, you can solve for x by isolating it on one side of the equation. The steps to find x are as follows:
1) Write out the expression for the sum of the arc measures:
Arc QP + Arc PR + Arc QR = 360°
2) Replace the arc measures with their corresponding expressions:
(8x - 10) + (6x) + (10x + 10) = 360°
3) Simplify the equation by combining like terms:
24x = 360°
4) Solve for x by isolating it:
x = 360° / 24
x = 15°
Now that we have found the value of x, we can move on to part B of the question.
To determine if the triangle is equilateral, isosceles, or scalene, we need to find the measures of angles <P, <R, and <S. We can use the Inscribed Angles Theorem to find these angle measures. The Inscribed Angles Theorem states that the measure of an inscribed angle is half the measure of its intercepted arc.
1) Plug in the value of x we found (x = 15°) into each of the arc measures:
Arc QP = 8x - 10 = 8(15) - 10 = 110°
Arc PR = 6x = 6(15) = 90°
Arc QR = 10x + 10 = 10(15) + 10 = 160°
2) Apply the Inscribed Angles Theorem to find <P, <R, and <S:
<P = Arc QR / 2 = 160° / 2 = 80°
<R = Arc QP / 2 = 110° / 2 = 55°
<S = Arc PR / 2 = 90° / 2 = 45°
Now, we can use the angle measures we found above to determine what type of triangle we have:
In an equilateral triangle, all three angles are equal. Since <P = 80°, <R = 55°, and <S = 45°, the triangle is not equilateral.
In an isosceles triangle, two angles are equal. Since none of the angles are equal in this case, the triangle is not isosceles.
In a scalene triangle, all three angles are different. Since <P = 80°, <R = 55°, and <S = 45°, the triangle is scalene.
Therefore, based on the angle measures, we can conclude that the triangle is scalene.