Three points A, B and C have coordinates A (8, 17), B (15, 10) and C (−2, −7) 7 (a) Show that angle ABC is a right angle - AQA - A-Level Maths Mechanics - Question 7 - 2018 - Paper 1

To show that angle ABC is a right angle, we can use the coordinates of points A, B, and C to determine the slopes of lines AB and BC, or alternatively find the lengths of the sides and use the Pythagorean theorem.

1. Finding the lengths of sides:

   • Distance AB:
     $AB = ext{sqrt}((15 - 8)^2 + (10 - 17)^2) = ext{sqrt}(7^2 + (-7)^2) = ext{sqrt}(49 + 49) = ext{sqrt}(98) = 7 ext{sqrt}(2)$
   • Distance BC:
     $BC = ext{sqrt}((-2 - 15)^2 + (-7 - 10)^2) = ext{sqrt}((-17)^2 + (-17)^2) = ext{sqrt}(289 + 289) = ext{sqrt}(578) = 17 ext{sqrt}(2)$
   • Distance AC:
     $AC = ext{sqrt}((-2 - 8)^2 + (-7 - 17)^2) = ext{sqrt}((-10)^2 + (-24)^2) = ext{sqrt}(100 + 576) = ext{sqrt}(676) = 26$

2. Verifying Pythagorean Theorem:

   • According to the theorem:
     $AB^2 + BC^2 = AC^2$
   • Calculating:
     $(7 ext{sqrt}(2))^2 + (17 ext{sqrt}(2))^2 = 676$
   • Thus:
     $98 + 578 = 676$
   • Therefore, angle ABC is a right angle.