The points \(A(2,\,1)\), \(B(6,\,4)\) and \(C(3,\,8)\) are joined to form triangle ABC. Which description fits this triangle?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → two sides equal, with one right angle
The sides are found with the distance formula. Two equal sides make the triangle isosceles, and if the square of the longest side equals the sum of the squares of the other two, it is right-angled (converse of Pythagoras’ theorem).
\(AB^2=(6-2)^2+(4-1)^2=16+9=25\), so \(AB=5\)
\(BC^2=(3-6)^2+(8-4)^2=9+16=25\), so \(BC=5\)
\(CA^2=(2-3)^2+(1-8)^2=1+49=50\), so \(CA=5\sqrt{2}\)
\(AB=BC\), so the triangle is isosceles (not equilateral, since \(CA\) is longer)
\(AB^2+BC^2=25+25=50=CA^2\), so the angle at B is \(90^\circ\)
