Question

Find the lengths of the sides of the triangle PQR. P(0, 1, 5), Q(2, 3, 4), R(2, −3, 1) |PQ| = Correct: Your answer is correct. |QR| = Correct: Your answer is correct. |RP| = Correct: Your answer is correct. Is it a right triangle? Yes No Is it an isosceles triangle? Yes No

Answer 1

1. The values of |PQ|, |QR| and |RP| are 3, 3√5 and 6 respectively.

2. No.

3. No.

Explanation:

The vertices of given triangle are P(0, 1, 5), Q(2, 3, 4), R(2, −3, 1).

Distance formula:

`D=√((x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2)`

Using distance formula we get

`|PQ|=√((2-0)^2+(3-1)^2+(4-5)^2)=√(9)=3`

`|QR|=√((2-2)^2+(-3-3)^2+(1-4)^2)=√(45)=3√(5)`

`|RP|=√((0-2)^2+(1-(-3))^2+(5-1)^2)=√(36)=6`

The values of |PQ|, |QR| and |RP| are 3, 3√5 and 6 respectively.

In a right angled triangle the sum of squares of two small sides is equal to the square of third side.

`(3)^2+(3√(5))^2=54\\eq 6^2`

Therefore PQR is not a right angled triangle.

In an isosceles triangle, the length of two sides are equal.

The measure of all sides are different, therefore PQR is not an isosceles triangle.