In the $xy$-coordinate plane, triangle $RST$ is equilateral. Points $R$ and $T$ have coordinates $(0,2)$ and $(1,0)$, respectively.

Quantity A |
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | Quantity B |

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | ||

The perimeter of triangle $RST$ | $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | $3√5$ |

- Quantity A is greater.
- Quantity B is greater.
- The two quantities are equal.
- The relationship cannot be determined from the information given.

So, you were trying to be a good test taker and practice for the GRE with PowerPrep online. Buuuut then you had some questions about the quant section—specifically question 5 of the second Quantitative section of Practice Test 1. Those questions testing our knowledge of **Coordinate Geometry** can be kind of tricky, but never fear, PrepScholar has got your back!

## Survey the Question

Let’s search the problem for clues as to what it will be testing, as this will help shift our minds to think about what type of math knowledge we’ll use to solve this question. Pay attention to any words that sound math-specific and anything special about what the numbers look like, and mark them on our paper.

Since the question mentions the **$xy$-coordinate plane** and gives us **points** and their coordinates, we will likely rely on what we know about **Coordinate Geometry**. Let’s keep what we’ve learned about this skill at the tip of our minds as we approach this question.

## What Do We Know?

Let’s carefully read through the question and make a list of the things that we know.

- We have an equilateral triangle $RST$ in the $xy$-coordinate plane
- Points $R$ and $T$ have coordinates $(0,2)$ and $(1,0)$, respectively
- We want to compare the perimeter of triangle $RST$ to the value $3√5$

## Develop a Plan

We want to find the perimeter of equilateral triangle $RST$. We know that the **perimeter** of a triangle is the sum of its three sides. We also know that, in an equilateral triangle, the three sides are equal. So if we can find the length of one side,

then we can multiply that length by $3$ to get the perimeter of the triangle.

From our knowledge of Coordinate Geometry, we know that we can calculate the distance between any two points $(x_1,y_1)$ and $(x_2,y_2)$ using the distance formula, which is the square root of the sum of the squares of the differences between the $x$-values and the differences between the $y$-values of the two points.

$\Distance \between \Two \Points =$$√{(\;\;x_{\;\;2}\;\;-\;\;x_{\;\;1}\;\;)^{\;\;2}\;\;+\;\;(\;\;y_{\;\;2}\;\;-\;\;y_{\;\;1}\;\;)^{\;\;2}$

Since we know the coordinates of points $R$ and $T$, let’s plug their $x$- and $y$-values into the distance formula. Then after we get the length of one side of triangle $RST$, we can multiply this value by $3$ to get the perimeter of the triangle.

## Solve the Question

Using the distance formula to find the length of side $RT$ using points $(0,2)$ and $(1,0)$, we get:

$\Length \of RT$ | $=$ | $√{(\;\;2\;\;-\;\;0\;\;)^{\;\;2}\;\;+\;\;(\;\;0\;\;-\;\;1\;\;)^{\;\;2}$ |

$ $ | $ $ | |

$\Length \of RT$ | $=$ | $√{(\;\;2\;\;)^{\;\;2}\;\;+\;\;(\;-\;1\;\;)^{\;\;2}$ |

$ $ | $ $ | |

$\Length \of RT$ | $=$ | $√{4+1}$ |

$ $ | $ $ | |

$\Length \of RT$ | $=$ | $√5$ |

Next, we know that if the length of one side of an equilateral is $√5$, then the perimeter must be three times as much, or $3√5$ , which is the same as Quantity B.

**The correct answer is C, the two quantities are equal**.

## What Did We Learn

There won’t be too many formulas that we should commit to memory in preparation for the GRE quant section, but the distance formula is definitely one of them. So let’s write it a few times on our paper right now so that it’ll better stick in our minds.

$\Distance \between \Two \Points =$$√{(\;\;x_{\;\;2}\;\;-\;\;x_{\;\;1}\;\;)^{\;\;2}\;\;+\;\;(\;\;y_{\;\;2}\;\;-\;\;y_{\;\;1}\;\;)^{\;\;2}$

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