Question

Use the distance formula and the Pythagorean Theorem to determine if thefollowing three coordinate points form a right triangle:A: (-2, 1)B: (0,4)C: (4,2)

Answer 1

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;

`|d|=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2_{}_{}_{}}`

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.

`|AB|=\sqrt[]{(0+2)^2+(4-1)^2}`
`\begin{gathered} =\sqrt[]{2^2+3^2} \\ =\sqrt[]{4+9} \\ =\sqrt[]{13} \end{gathered}`

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.

`\begin{gathered} |BC|=\sqrt[]{(4-0)^2+(2-4)^2} \\ \\ =\sqrt[]{4^2+(-2)^2} \\ =\sqrt[]{16+4} \\ =\sqrt[]{20} \end{gathered}`

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.

`\begin{gathered} |CA|=\sqrt[]{(-2-4)^2+(1-2)^2} \\ \\ =\sqrt[]{(-6)^2+(1)^2} \\ \\ =\sqrt[]{36+1} \\ =\sqrt[]{37} \end{gathered}`

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.