Respuesta :
It is NOT a right triangle.
STEP - BY - STEP EXPLANATION
What to find?
• Side AB
,• Side BC
,• Side CA
,• Determine whether the triangle satisfies the ,pythagoras theorem,.
Given:
• A: (-2, 1)
,• B: (0,4)
,• C: (4,2)
We will follow the steps below to solve the given problem.
Step 1
Recall the distance formula.
That is;
[tex]|d|=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2_{}_{}_{}}[/tex]Step 2
Find the distance AB.
Given the coordinates A: (-2, 1) and B: (0,4)
x₁= -2 y₁=1 x₂= 0 y₂=4
Substitute the values into the given formula and evaluate.
[tex]|AB|=\sqrt[]{(0+2)^2+(4-1)^2}[/tex][tex]\begin{gathered} =\sqrt[]{2^2+3^2} \\ =\sqrt[]{4+9} \\ =\sqrt[]{13} \end{gathered}[/tex]Hence, side AB =√13
Step 2
Determine side length BC.
B: (0,4) and C: (4,2)
x₁= 0 y₁=4 x₂= 4 y₂=2
Substitute the values into the formula and simplify.
[tex]\begin{gathered} |BC|=\sqrt[]{(4-0)^2+(2-4)^2} \\ \\ =\sqrt[]{4^2+(-2)^2} \\ =\sqrt[]{16+4} \\ =\sqrt[]{20} \end{gathered}[/tex]Hence, BC = √20
Step 3
Solve for length CA
C: (4,2) and A: (-2, 1)
x₁= 4 y₁=2 x₂= -2 y₂=1
Substitute the values into the formula and simplify.
[tex]\begin{gathered} |CA|=\sqrt[]{(-2-4)^2+(1-2)^2} \\ \\ =\sqrt[]{(-6)^2+(1)^2} \\ \\ =\sqrt[]{36+1} \\ =\sqrt[]{37} \end{gathered}[/tex]Side length CA = √37
Step 4
Check if it satisfies the Pythagoreans theorem.
Using Pythagoras theorem;
opposite² + adjacent² = hypotenuse²
Let hypotenuse = √37 opposite =√20 and adjacent =√13
Substitute the values at the left-hand side and simplify to see if the final result gives a value at the right-hand side.
That is;
(√20)² + (√13)² = 20 + 13 = 33 = hypotenuse²
⇒hypotenuse = √33
Clearly, we can see that it gives √33 which is not equal to √37.
Hence, it does not satifies the Pythagorean Theorem .
Therefore, this follows that it is NOT a right triangle.
