Question 6 of 70intermediate🔍 AnalyzeLong Answer4 marks

Without using the Pythagoras theorem, show that the points (4,4)(4, 4), (3,5)(3, 5) and (−1,−1)(-1, -1) are the vertices of a right angled triangle.

Correct Answer

Let the given points be A(4,4)A(4, 4), B(3,5)B(3, 5), and C(−1,−1)C(-1, -1).

Finding slopes of the sides:

Slope of AB:

m1=y2−y1x2−x1=5−43−4=1−1=−1m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 4}{3 - 4} = \frac{1}{-1} = -1

Slope of BC:

m2=−1−5−1−3=−6−4=32m_2 = \frac{-1 - 5}{-1 - 3} = \frac{-6}{-4} = \frac{3}{2}

Slope of CA:

m3=4−(−1)4−(−1)=55=1m_3 = \frac{4 - (-1)}{4 - (-1)} = \frac{5}{5} = 1

Checking for perpendicular lines:

Product of slopes of AB and CA:

m1×m3=(−1)×1=−1m_1 \times m_3 = (-1) \times 1 = -1

Since the product of slopes of AB and CA equals −1-1, these two sides are perpendicular to each other.

Therefore, the angle at vertex A is 90°90°, which means the triangle formed by points A(4,4)A(4, 4), B(3,5)B(3, 5), and C(−1,−1)C(-1, -1) is a right-angled triangle with the right angle at A.

Exercise: EXERCISE 9.1 | Q: 6 | (Page 8)
For More Understanding

Explanation

This question tests the concept that two lines are perpendicular if and only if the product of their slopes equals −1-1. The textbook covers slope calculations in the context of finding slopes of lines passing through given points. By calculating slopes of all three sides and checking which pair multiplies to −1-1, we identify the perpendicular sides without using distance formula or Pythagoras theorem.

Solution Steps

  1. Step 1: Label the points as A(4,4)A(4, 4), B(3,5)B(3, 5), and C(−1,−1)C(-1, -1).

  2. Step 2: Calculate slope of AB using formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}, getting m1=−1m_1 = -1.

  3. Step 3: Calculate slope of BC, getting m2=32m_2 = \frac{3}{2}.

  4. Step 4: Calculate slope of CA, getting m3=1m_3 = 1.

  5. Step 5: Check product of slopes: m1×m3=−1m_1 \times m_3 = -1, proving AB⊥CAAB \perp CA.

  6. Step 6: Conclude triangle is right-angled at vertex A.

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