show that triangle PQR with P(-2,10, Q(1,5) and R(5,2) is isosceles.
Find out the lengths of each side using the distance formula:
D=√((x2-x1)²+(y2-y1)²)
If two sides are identical in length, the triangle is isosceles.
If all three have the same length, then it is equilateral.
To show that triangle PQR is isosceles, we need to prove that at least two sides of the triangle have equal lengths.
The distance between two points (x1, y1, z1) and (x2, y2, z2) in three-dimensional space can be calculated using the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
Let's calculate the distances between the three pairs of points:
PQ:
d(PQ) = √((1 - (-2))^2 + (5 - 10)^2 + (0 - 10)^2)
= √(3^2 + (-5)^2 + (-10)^2)
= √(9 + 25 + 100)
= √(134)
QR:
d(QR) = √((5 - 1)^2 + (2 - 5)^2 + (0 - 10)^2)
= √(4^2 + (-3)^2 + (-10)^2)
= √(16 + 9 + 100)
= √(125)
RP:
d(RP) = √((-2 - 5)^2 + (10 - 2)^2 + (10 - 0)^2)
= √((-7)^2 + 8^2 + 10^2)
= √(49 + 64 + 100)
= √(213)
After calculating the distances, we find that:
d(PQ) = √(134)
d(QR) = √(125)
d(RP) = √(213)
Since the square roots of these distances are not equal, it means that none of the sides of triangle PQR have equal lengths. Therefore, triangle PQR is not isosceles.
Note: In order to prove that a triangle is isosceles, we need to show that two sides have equal lengths, or that two angles have equal measures.