Question

Let A = (−2, 6), B = (1, 0), and C = (5, 2). Prove that △ABC is a right-angled triangle.Let u = AB, v = BC, and w = AC. We must show that u · v, u · w, or v · w is zero in order to show that one of these pairs is orthogonal.u · v = Incorrect: Your answer is incorrect.u · w = Incorrect: Your answer is incorrect.v · w = Incorrect: Your answer is incorrect.

Answer 1

Answer:

Step-by-step explanation:

In a triangle ABC, given A= (-2,6),

B = (1,0) and C = (5,2)

If u = AB, v = BC, w = AC

To show that the triangle is a right angled triangle, we must show that the dot product of one of the pairs is zero.

Since u = AB

u = AB = B-A

u = AB = (1,0) - (-2,6)

u = [(1-(-2), 0-6]

u = (3, -6)

Similarly, v = BC = C-B

v = BC = (5,2) - (1,0)

v = [(5-1), (2-0)]

v = (4, 2)

Also for w:

w = AC = C - A

w = (5, 2) - (2, -6)

w = [(5-2), (2-(-6)]

w = (3, 8)

To show that the triangle is a right angled triangle, the dot product of one of any of the pairs must be zero as shown:

u.v = (3, -6) • (4, 2)

u.v = (3)(4) + (-6)(2)

u.v = 12-12

u.v = 0

i.e AB.BC = 0

This shows that length AB and BC are perpendicular to each other i.e the angle between them is 90° and since a right angled triangle has one of its angle to be 90°, it shows that the ∆ABC is a right angled triangle.

Answer 2

The equation of the dot product of the vector representation of two of

the sides of the triangle ΔABC is zero, if the sides are perpendicular.

Response:

• The sides of the triangle AB and BC are orthogonal, given that u·v = 0, therefore, ΔABC is a right triangle.

Method used to prove that the triangle is a right-angled triangle

The given parameters are;

The length of side AB = u

The length of side BC = v

The length of side AC = w

The given coordinates are;

A = (-2, 6), B = (1, 0), and C = (5, 2)

Representing the given points as vectors gives;

A = -2·i + 6·j

B = 1·i + 0·j

C = 5·i + 2·j

Vector representation of AB = 1·i - (-2·i) + 0 - 6·j = 3·i - 6·j

• Therefore, AB = (3, -6) = u

Vector representation of BC = 5·i - 1·i + 2·j - 0 = 4·i + 2·j

• Therefore BC = (4, 2) = v

Vector representation of AC = 5·i - (-2·i) + 2·j - 6·j = 7·i - 4·j

• Therefore, length of AC = (7, -4) = w

Given that ΔABC is a right angled-triangle, we have;

• Either, AB ⊥ BC, or AC ⊥ BC, or AB ⊥ BC

The dot product of perpendicular lines = 0, therefore;

Therefore, we have;

• Either u·v = 0, or u·w = 0, or v·w = 0

The dot product, u·v = (3, -6)·(4, 2) = 3 × 4 - 6 × 2 = 0

u·w = (3, -6)·(7, -4) = 3 × 7 + 6 × 4 =45

v·w = (4, 2)·(7, -4) = 28 - 14 = 14

Therefore;

Given that u·v = 0, the sides AB and BC are orthogonal and the triangle ΔABC is a right angled-triangle.

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