# Determine the following by using the method of vectors: The three vertices of a triangle X(4,1,7), Y(-2,1,1), Z(-3,5,-6), is it a right angle triangleIf it is not a right angle triangle, state what...

Determine the following by using the method of vectors: The three vertices of a triangle X(4,1,7), Y(-2,1,1), Z(-3,5,-6), is it a right angle triangle

If it is not a right angle triangle, state what type of triangle it is and the lengths of the triangle.

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### 1 Answer

You need to write the vectors representing the sides of triangle:

`bar(XY) = (-2-4)*bari + (1-1)*barj + (1-7)bark`

`bar(XY) = (-6)*bari + (-6)bark`

`bar(XZ) = (-3-4)*bari + (5-1)*barj + (-6-7)bark`

`bar(XZ) = (-7)*bari + (4)*barj + (-13)bark`

You need to check if the vectors XY and XZ are perpendicular. You may verify this relation between the vectors using the dot product such that:

`bar(XY)*bar(XZ) = |bar(XY)|*|bar(XZ)|*cos alpha`

`alpha` denotes the angle between `bar(XY)` and `bar(XZ).`

`bar(XY)*bar(XZ) = [(-6)*bari + (-6)bark]*[(-7)*bari + (4)*barj + (-13)bark]`

`bar(XY)*bar(XZ) = (-6)*(-7) + 0*4 + (-6)*(-13)`

`bar(XY)*bar(XZ) = 42 + 78 =gt bar(XY)*bar(XZ) = 120`

`|bar(XY)| = sqrt(36 + 36) =gt |bar(XY)| =6sqrt2`

`|bar(XZ)| = sqrt((-7)^2 + 4^2 + (-13)^2)`

`|bar(XZ)| = sqrt(49 +16 + 169)`

`|bar(XZ)| = sqrt(234)`

`cos alpha = (bar(XY)*bar(XZ) )/(|bar(XY)|*|bar(XZ)|)(` `cos alpha =120/(6sqrt2*sqrt(234))=gtcos alpha =20/sqrt 468=gt cos alpha ~~20/21.63 =gt cos alpha ~~0.924`

**Since `cos alpha != 0` => the angle between vectors `bar(XY) ` and `bar(XZ)` is `!=` 90 degrees => the triangle XYZ is not right triangle.**