The points P (2, −1), Q (−4, −1) and R (−1, 3�√3 − 1) are joined to form a triangle. Prove that triangle
PQR is equilateral.
I get that you use the distance formula to find the distance between these three points, which will prove that PQR is equilateral.
I have found PQ (√36 = 6) (please correct me if i'm wrong), but i'm having trouble finding the distance between PR and QR because of '3�√3 − 1'. I'm not sure if I should add, whether to convert it to decimal beforehand, etc.
I would appreciate any help given. Thanks in advanced.
Plug in the formula before you get worried. The -1 goes away:
(-1-2)^2 + (3√3 − 1+1)^2
= 3^2 + (3√3)^2
= 9+27
= 36
And in any case, forget the decimal stuff -- that's just approximate. √3 is exact.
To prove that triangle PQR is equilateral, we need to show that all sides of the triangle PQR are equal in length.
Let's find the distances between the three points using the distance formula. The distance formula states that the distance between two points (x1, y1) and (x2, y2) is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Calculating the distances:
1. PQ: Given P (2, -1) and Q (-4, -1).
d(PQ) = √((-4 - 2)^2 + (-1 - (-1))^2)
= √((-6)^2 + (0)^2)
= √(36)
= 6
2. PR: Given P (2, -1) and R (-1, 3√3 - 1).
d(PR) = √((-1 - 2)^2 + (3√3 - 1 - (-1))^2)
= √((-3)^2 + (3√3 - 2)^2)
= √(9 + 9(3)(√3) - 12(√3) + 4)
= √(9 + 27√3 - 6√3 + 4)
= √(13 + 21√3)
3. QR: Given Q (-4, -1) and R (-1, 3√3 - 1).
d(QR) = √((-1 - (-4))^2 + (3√3 - 1 - (-1))^2)
= √(3^2 + (3√3 - 2)^2)
= √(9 + 9(3)(√3) - 12(√3) + 4)
= √(9 + 27√3 - 6√3 + 4)
= √(13 + 21√3)
Now, to prove that the triangle PQR is equilateral, we need to show that PQ = PR = QR.
We know that PQ = 6.
To find out if PR = QR, we need to evaluate √(13 + 21√3) and determine if it is equal to 6.
To simplify √(13 + 21√3), we can use a calculator or convert it to decimal form.
Calculating √(13 + 21√3) using a calculator, we get approximately 9.392.
Since PQ = 6, and PR ≈ 9.392 ≠ 6, we can conclude that the triangle PQR is not equilateral.
Therefore, the given statement "Prove that triangle PQR is equilateral" is false.