Question

On a coordinate plane, triangles R S T and L M N are shown. Triangle R S T has points (5, 5), (2, 1), and (1, 3). Triangle L M N has points (0, negative 1), (2, negative 4), and (negative 2, negative 3). How does the area of triangle RST compare to the area of triangle LMN? The area of △RST is 2 square units less than the area of △LMN. The area of △RST is equal to the area of △LMN. The area of △RST is 2 square units greater than the area of △LMN. The area of △RST is 4 square units greater than the area of △LMN.

Answer 1

Answer:

The answer should be "B" (The area of △RST is equal to the area of △LMN.)

Answer 2

The area of ΔRST is equal to the area of ΔLMN.

How to find the area of a triangle from its vertices ?

We know that, if the vertices of a triangle are $(x_{1},y_{1}), (x_{2} ,y_{2} ), (x_{3} ,y_{3} )$,

then the area of the triangle is $\frac{1}{2}$ | $x_{1} (y_{2} -y_{3} )+x_{2} (y_{3} -y_{1} )+x_{3} (y_{1} -y_{2} )$ |

What is the result of comparing area of two given triangles ?

In ΔRST, Vertices are R=(5,5), S=(2,1), T=(1,3)

In ΔLMN, Vertices are L=(0,-1), M=(2,-4), N=(-2,-3)

Area of ΔRST =  $\frac{1}{2}$ | $x_{1} (y_{2} -y_{3} )+x_{2} (y_{3} -y_{1} )+x_{3} (y_{1} -y_{2} )$ |

                      =  $\frac{1}{2}$ | 5(1-3)+2(3-5)+1(5-1) |

                      =  $\frac{1}{2}$ | 5×(-2)+2×(-2)+4 |

                      =  $\frac{1}{2}$ | -10-4+4 |

                      =  $\frac{1}{2}$ | -10 |

Removing the absolute sign, we get,

Area of ΔRST =  $\frac{1}{2}$ ×10 = 5 square unit

Again, Area of ΔLMN =  $\frac{1}{2}$ | $x_{1} (y_{2} -y_{3} )+x_{2} (y_{3} -y_{1} )+x_{3} (y_{1} -y_{2} )$ |

                                   =  $\frac{1}{2}$ | 0(-4+3)+2(-3+1)-2(-1+4) |

                                   =  $\frac{1}{2}$ | 0+2(-2)-2(3) |

                                   =  $\frac{1}{2}$ | -4-6 |

                                   =  $\frac{1}{2}$ | -10 |

Removing the absolute sign, we get,

Area of ΔLMN =  $\frac{1}{2}$ ��10 = 5 square unit

Hence, we can say that, the area of ΔRST = the area of ΔLMN

Learn more about area of a triangle from its vertices here :

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