Question

if arc lm = (8x - 56)° and arc np = (5x + 22)°, find arc lp.
answer

Explanation:

Step1: Use the property of arcs in a circle

In a circle, if two chords are equidistant from the center, then their arcs are equal. Here, arc \(LM\) and arc \(NP\) are equal. So, \(8x - 56=5x + 22\).

Step2: Solve the equation for \(x\)

Subtract \(5x\) from both sides: \(8x-5x - 56=5x-5x + 22\), which gives \(3x-56 = 22\).
Add \(56\) to both sides: \(3x-56 + 56=22 + 56\), so \(3x=78\).
Divide both sides by \(3\): \(x=\frac{78}{3}=26\).

Step3: Find the measure of arc \(LM\) or \(NP\)

Substitute \(x = 26\) into \(8x-56\): \(8\times26-56=208 - 56=152^{\circ}\).

Step4: Find the measure of arc \(LP\)

The sum of arcs in a circle is \(360^{\circ}\). We know arc \(MN = 39^{\circ}\), and arc \(LM=\)arc \(NP = 152^{\circ}\).
Let arc \(LP=y\). Then \(y+152 + 152+39=360\).
Simplify the left - hand side: \(y+343 = 360\).
Subtract \(343\) from both sides: \(y=360 - 343=17^{\circ}\).

Answer:

\(17^{\circ}\)