Verify the following: (0, 7, –10), (1, 6, – 6) and (4, 9, – 6) are the vertices of an isosceles triangle.

Asked by Abhisek | 11 months ago |  81

##### Solution :-

(0, 7, –10), (1, 6, – 6) and (4, 9, – 6) are the vertices of an isosceles triangle.

Let us consider the points be

P(0, 7, –10), Q(1, 6, – 6) and R(4, 9, – 6)

If any 2 sides are equal, hence it will be an isosceles triangle

So firstly let us calculate the distance of PQ, QR

Calculating PQ

P ≡ (0, 7, – 10) and Q ≡ (1, 6, – 6)

By using the formula,

Distance PQ =

$$\sqrt{(x^2 – x^1)^2 + (y^2 – y^1)^2 + (z^2 – z^1)^2}$$

So here,

x1 = 0, y1 = 7, z1 = – 10

x2 = 1, y2 = 6, z2 = – 6

Distance PQ =

$$\sqrt{(1 – 0)^2 + (6 – 7)^2 + (-6 – (-10))^2}$$

$$\sqrt{(1)^2 + (-1)^2 + (4)^2}$$

$$\sqrt{1 + 1 + 16}$$

$$\sqrt{18}$$

Calculating QR

Q ≡ (1, 6, – 6) and R ≡ (4, 9, – 6)

By using the formula,

Distance QR =

$$\sqrt{(x^2 – x^1)^2 + (y^2 – y^1)^2 + (z^2 – z^1)^2}$$

So here,

x1 = 1, y1 = 6, z1 = – 6

x2 = 4, y2 = 9, z2 = – 6

Distance QR =

$$\sqrt{(4 – 1)^2 + (9 – 6)^2 + (-6 – (-6))^2}$$

$$\sqrt{(3)^2 + (3)^2 + (-6+6)^2}$$

$$\sqrt{9 + 9 + 0}$$

$$\sqrt{18}$$

Hence, PQ = QR

18 = 18

2 sides are equal

PQR is an isosceles triangle.

Answered by Pragya Singh | 11 months ago

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