Question

the length of an arc of a circle is 13.9 feet. the arc subtends a central angle that measures 131.7°. what is the approximate circumference of the circle?
31.2 feet
38.0 feet
111.7 feet
119.4 feet
in parallelogram mnpq,

• the diagonals intersect at point g,
• mg = (4x - 6) cm,
• mp = (10x - 22) cm,

what is gp?
5 cm
10 cm
14 cm
28 cm

Explanation:

First Question (Arc Length and Circumference)

Step1: Recall Arc Length Formula

The arc length \( s \) of a circle is related to the central angle \( \theta \) (in degrees) and the circumference \( C \) by the formula \( s=\frac{\theta}{360^\circ}\times C \).

Step2: Rearrange for Circumference

We can rearrange the formula to solve for \( C \): \( C = s\times\frac{360^\circ}{\theta} \).

Step3: Substitute Values

Given \( s = 13.9 \) feet and \( \theta=131.7^\circ \), substitute into the formula:
\( C=13.9\times\frac{360}{131.7} \).

Step4: Calculate

First, calculate \( \frac{360}{131.7}\approx2.733 \). Then, \( 13.9\times2.733\approx38.0 \) feet.

Second Question (Parallelogram Diagonals)

Step1: Recall Parallelogram Diagonal Property

In a parallelogram, the diagonals bisect each other. So, \( MG = GP \) and \( MP=MG + GP = 2\times MG \).

Step2: Set Up Equation

Given \( MG=(4x - 6) \) cm and \( MP=(10x - 22) \) cm, we have \( 10x - 22 = 2(4x - 6) \).

Step3: Solve for \( x \)

Expand the right - hand side: \( 10x - 22 = 8x-12 \).
Subtract \( 8x \) from both sides: \( 10x - 8x-22=8x - 8x-12 \), which gives \( 2x-22=-12 \).
Add 22 to both sides: \( 2x-22 + 22=-12 + 22 \), so \( 2x = 10 \).
Divide by 2: \( x = 5 \).

Step4: Find \( MG \) and Then \( GP \)

Substitute \( x = 5 \) into \( MG = 4x-6 \): \( MG=4(5)-6=20 - 6 = 14 \) cm.
Since \( MG = GP \) (diagonals bisect each other), \( GP = 14 \) cm.

Answer:

38.0 feet