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How can we determine whether the triangles $ABC$ and $DEF$ are rectangles? We have $A(-6,5),B(-3,3),C(1,9),D(1,3),E(5,1),F(11,10)$.

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At least in spanish, a right angle triangle is called "triángulo rectángulo"="rectangle-triangle", for the pretty obvious reason. Most probably this what the OP means to ask: are those triangles right-angled ones? –  DonAntonio Nov 20 '12 at 17:38

2 Answers 2

Triangles have three sides and rectangles four, so triangles are never rectangles. Maybe you mean to ask whether they are right triangles? If so, check whether the side lengths match the Pythagorean theorem: $a^2+b^2=c^2$. To get the length of a side, if it goes from $(x_1,y_1)$ to $(x_2,y_2)$ the length is $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

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You must have two sides perpendicular to each other, and two lines (or segments of lines) are perpendiculars iff their slopes' product is $\,-1\,$ , so do as follows:

$$(1)\;\;\;\;\;\text{Calculate the three sides slopes}$$

$$(2)\;\;\;\;\;\text{Check whether the product of two of the above slopes is}\,\,-1$$

For example, for the first triangle:

$$m_{AB}=\frac{5-3}{-6+3}=-\frac{2}{3}\,\,,\,\,m_{AC}=\frac{5-9}{-6-1}=\frac{4}{7}\,\,,\,\,m_{BC}=\frac{3-9}{-3-1}=\frac{3}{2}$$

so $\,\Delta ABC\,$ is a right angled triangle (where is the right angle located?), etc..

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