## What is the circumference of…

## Solution:

We need to determine the circumference of the circle provided. Let’s sketch the diagram below.

We see from our diagram that we have a right triangle in a circle. We also see that side OX and side OZ represent legs of the right triangle, and they are also each a radius of the circle. Since we have a right triangle with two equal sides, we have a 45-45-90 right triangle. The sides of a 45-45-90 right triangle can be represented by s, s, and s√2, respectively. Let’s add the new information to our diagram.

**Statement One Alone: **

The perimeter of triangle OXZ is 20 + 10√2.

From our given diagram, we can express the perimeter of triangle OXZ as s + s + s√2.

Using the information from statement two we can create the following equation:

s + s + s√2 = 20 + 10√2

2s + s√2 = 20 + 10√2

s(2 + √2) = 10(2 + √2)

s = 10

From this equation, we see that s must equal 10 in order for 2s + s√2 to equal 20 + 10√2.

Since we know that s = 10, we also know that the radius of the circle is 10, and this is enough information to determine the circle’s circumference. Statement one alone is sufficient to answer the question. We can eliminate answer choices B, C, and E.

**Statement Two Alone: **

The length of arc XYZ is 5π.

To determine arc length we can use the following proportion:

central angle/360 = arc length/circumference

We have a central angle of 90 degrees, so we can use that value for the central angle.

90/360 = 5π/circumference

¼ = 5π/circumference

20π = circumference

Statement two alone is sufficient to answer the question.

**Answer: D**