Question

find the measure of ec.

Explanation:

Step1: Recall circle - arc sum property

The sum of the measures of the arcs of a circle is 360°.

Step2: Set up an equation

We know that \(5x + 7x+90^{\circ}\) plus the measure of the remaining arc (which we don't need to explicitly find) sums to 360°. But we can also note that the sum of the given arcs \(5x + 7x+90^{\circ}\) and the other arcs in the circle must satisfy the 360 - degree rule. In terms of the arcs related to our problem, we have \(5x+7x + 90^{\circ}=360^{\circ}\).
First, combine like - terms: \(12x+90^{\circ}=360^{\circ}\).
Then, subtract 90° from both sides: \(12x=360^{\circ}-90^{\circ}=270^{\circ}\).
Next, solve for \(x\): \(x = \frac{270^{\circ}}{12}=22.5^{\circ}\).

Step3: Find the measure of arc \(EC\)

The measure of arc \(EC\) is \(5x\). Substitute \(x = 22.5^{\circ}\) into \(5x\). So, the measure of arc \(EC=5\times22.5^{\circ}=112.5^{\circ}\).

Answer:

\(112.5^{\circ}\)