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Triangle XYZ is shown on the coordinate grid.



Which statements are true about triangle XYZ?


XY measures √26units.

YZ measures √56 units.

ZX measures √34 units.

XYZ is an isosceles triangle.

XYZ is a right triangle.

XYZ is a scalene triangle.

Triangle XYZ is shown on the coordinate gridWhich statements are true about triangle XYZXY measures 26unitsYZ measures 56 unitsZX measures 34 unitsXYZ is an iso class=

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Answer:

XY measures √26units.

XYZ is an isosceles triangle.

XYZ is a right triangle.

Step-by-step explanation:

The given triangle has vertices [tex]X(-1,5),Y(4,4),(-2,0)[/tex]


We now apply the distance formula to find the length of all sides to obtain;

[tex]|XY|=\sqrt{(4--1)^2+(4-5)^2}[/tex]


[tex]|XY|=\sqrt{(4+1)^2+(4-5)^2}[/tex]


[tex]|XY|=\sqrt{5^2+(-1)^2}[/tex]


[tex]|XY|=\sqrt{25+1}[/tex]


[tex]|XY|=\sqrt{26}[/tex]


The length of YZ is


[tex]|YZ|=\sqrt{(4--2)^2+(4-0)^2}[/tex]


[tex]|YZ|=\sqrt{(4+2)^2+(4-0)^2}[/tex]


[tex]|YZ|=\sqrt{6^2+(4)^2}[/tex]


[tex]|YZ|=\sqrt{36+16}[/tex]


[tex]|YZ|=\sqrt{52}[/tex]


The length of ZX is


[tex]|ZX|=\sqrt{(-2--1)^2+(0-5)^2}[/tex]


[tex]|ZX|=\sqrt{(-2+1)^2+(0-5)^2}[/tex]


[tex]|ZX|=\sqrt{(-1)^2+(-5)^2}[/tex]


[tex]|ZX|=\sqrt{1+25}[/tex]


[tex]|ZX|=\sqrt{26}[/tex]


Since

[tex]|XY|=\sqrt{26}=|ZX|[/tex]


The triangle is isosceles.


Slope of XZ

[tex]\frac{5-0}{-1--2}=\frac{5}{1} =5[/tex]


Slope of XY

[tex]=\frac{4-5}{4--1}=-\frac{1}{5}[/tex]


Since

[tex]\frac{-1}{5}\times 5=-1[/tex]


The triangle is a right angle triangle.

Solution:

Distance between two points [tex](x_{1},y_{1}), (x_{2},y_{2})[/tex] in two dimensional coordinate plane is given by: [tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2[/tex]

[tex]XY=\sqrt{(4+1)^2+(4-5)^2}\\\\ XY=\sqrt{25 +1}\\\\ XY=\sqrt{26}\\\\ YZ=\sqrt{(4+2)^2+(4-0)^2}\\\\ YZ=\sqrt{36 +16}\\\\ YZ=\sqrt{52}\\\\ YZ=2\sqrt{13}\\\\ XZ=\sqrt{(-2+1)^2+(5-0)^2}\\\\ XZ=\sqrt{1+25}\\\\ XZ=\sqrt{26}\\\\ Also, (XY)^2+(XZ)^2=(\sqrt{26})^2+(\sqrt{26})^2=52=(YZ)^2[/tex]

As, Converse of Pythagorean theorem states that if sum of squares of any two adjacent sides in a triangle is equal to Square of length of third side, then the triangle is right angled triangle.

The true statement about the triangle XYZ is

1. XY measures √26units.

2. XYZ is an isosceles triangle.

3. XYZ is a right triangle.

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