Question

Name the type of triangle PQR formed by the points `P(sqrt(2), sqrt(2)), Q(-sqrt(2), -sqrt(2))` and `R(-sqrt(6), sqrt(6))`.

Answer 1

Given: `P(sqrt(2), sqrt(2)), Q(-sqrt(2), -sqrt(2)), R(-sqrt(6), sqrt(6))`

Step-wise calculation:

Using the distance formula `PQ = sqrt((x_2 − x_1)^2 + (y_2 − y_1)^2)`.

1. `PQ^2 = (−sqrt(2) - sqrt(2))^2 + (-sqrt(2) - sqrt(2))^2`

= `(-2sqrt(2))^2 + (-2sqrt(2))^2`

= 8 + 8

= 16

⇒ PQ = 4

2. `PR^2 = (-sqrt(6) − sqrt(2))^2 + (sqrt(6) - sqrt(2))^2`

= `(6 + 2 + 2sqrt(12)) + (6 + 2 - 2sqrt(12))` 

= 16

⇒ PR = 4

3. `QR^2 = (-sqrt(6) + sqrt(2))^2 + (sqrt(6) + sqrt(2))^2`

= `(6 + 2 − 2sqrt(12)) + (6 + 2 + 2sqrt(12))` 

= 16

⇒ QR = 4

PQ = PR = QR = 4, so all three sides are equal.

Therefore, triangle PQR is an equilateral triangle each interior angle = 60°.