Question
The circle shown below has a triangle enclosed.
P
(8x - 10)° (6x)°
Q (10x + 10)° R
a. 13
b. 12
c. 67
d. 15
P
(8x - 10)° (6x)°
Q (10x + 10)° R
a. 13
b. 12
c. 67
d. 15
Answers
GPT-4o mini
To solve the problem, we first need to understand how the angles relate to each other within the triangle \(PQR\) inscribed in the circle.
Given the angles:
- \( \angle P = (8x - 10)^\circ \)
- \( \angle Q = (6x)^\circ \)
- \( \angle R = (10x + 10)^\circ \)
According to the triangle angle sum property, the sum of the interior angles of a triangle is always \(180^\circ\). Thus, we can set up the equation as follows:
\[
(8x - 10) + (6x) + (10x + 10) = 180
\]
Now, we can simplify the left side:
\[
8x - 10 + 6x + 10 + 10x = 180
\]
Combine like terms:
\[
(8x + 6x + 10x) + (-10 + 10) = 180
\]
\[
24x = 180
\]
Next, we can solve for \(x\):
\[
x = \frac{180}{24} = 7.5
\]
Now that we have \(x\), we can find the measures of the angles \(P\), \(Q\), and \(R\):
1. \( \angle P = 8x - 10 = 8(7.5) - 10 = 60 - 10 = 50^\circ\)
2. \( \angle Q = 6x = 6(7.5) = 45^\circ\)
3. \( \angle R = 10x + 10 = 10(7.5) + 10 = 75 + 10 = 85^\circ\)
Now we will double-check that these angles sum to \(180^\circ\):
\[
50 + 45 + 85 = 180
\]
They do indeed sum to \(180^\circ\).
Assuming the question asked for value of \(x\):
- The possible options provided \(a. 13, b. 12, c. 67, d. 15\) do not contain the value \(7.5\).
If you intended to ask about one of the individual angles instead (like \( \angle Q = 45^\circ \)), please clarify as options don't seem to correspond to anything in our calculations.
Given the angles:
- \( \angle P = (8x - 10)^\circ \)
- \( \angle Q = (6x)^\circ \)
- \( \angle R = (10x + 10)^\circ \)
According to the triangle angle sum property, the sum of the interior angles of a triangle is always \(180^\circ\). Thus, we can set up the equation as follows:
\[
(8x - 10) + (6x) + (10x + 10) = 180
\]
Now, we can simplify the left side:
\[
8x - 10 + 6x + 10 + 10x = 180
\]
Combine like terms:
\[
(8x + 6x + 10x) + (-10 + 10) = 180
\]
\[
24x = 180
\]
Next, we can solve for \(x\):
\[
x = \frac{180}{24} = 7.5
\]
Now that we have \(x\), we can find the measures of the angles \(P\), \(Q\), and \(R\):
1. \( \angle P = 8x - 10 = 8(7.5) - 10 = 60 - 10 = 50^\circ\)
2. \( \angle Q = 6x = 6(7.5) = 45^\circ\)
3. \( \angle R = 10x + 10 = 10(7.5) + 10 = 75 + 10 = 85^\circ\)
Now we will double-check that these angles sum to \(180^\circ\):
\[
50 + 45 + 85 = 180
\]
They do indeed sum to \(180^\circ\).
Assuming the question asked for value of \(x\):
- The possible options provided \(a. 13, b. 12, c. 67, d. 15\) do not contain the value \(7.5\).
If you intended to ask about one of the individual angles instead (like \( \angle Q = 45^\circ \)), please clarify as options don't seem to correspond to anything in our calculations.