Do the points (3, 2), (–2, –3) and (2, 3) form a triangle? If so, name the type of triangle formed.
Step-by-Step Solution
Key Concept: Use the slope (or area) test to check whether three points are collinear. If they are not collinear, they form a triangle. Then apply the distance formula to find the lengths of the sides and use the Pythagoras theorem to identify the type of triangle (right‑angled, isosceles, equilateral, etc.).
1. Check collinearity
- Let the points be \(A(3,2)\), \(B(-2,-3)\) and \(C(2,3)\).
- Slope of \(AB\):
$$m_{AB}=\frac{-3-2}{-2-3}=\frac{-5}{-5}=1$$
- Slope of \(AC\):
$$m_{AC}=\frac{3-2}{2-3}=\frac{1}{-1}=-1$$
- Since \(m_{AB}
eq m_{AC}\), the points are not collinear. Hence they form a triangle.
2. Find the lengths of the sides using the distance formula
- \(AB\):
$$AB=\sqrt{(3-(-2))^{2}+(2-(-3))^{2}}=\sqrt{5^{2}+5^{2}}=\sqrt{50}$$
- \(BC\):
$$BC=\sqrt{(-2-2)^{2}+(-3-3)^{2}}=\sqrt{(-4)^{2}+(-6)^{2}}=\sqrt{16+36}=\sqrt{52}$$
- \(AC\):
$$AC=\sqrt{(3-2)^{2}+(2-3)^{2}}=\sqrt{1^{2}+(-1)^{2}}=\sqrt{2}$$
3. Identify the type of triangle
- Compute the squares of the sides: \(AB^{2}=50\), \(BC^{2}=52\), \(AC^{2}=2\).
- The largest side is \(BC\). Check Pythagoras theorem: \(AB^{2}+AC^{2}=50+2=52=BC^{2}\).
- Since the sum of the squares of the two smaller sides equals the square of the largest side, the triangle satisfies the condition for a right‑angled triangle.
- The right angle is at point \(A(3,2)\) because the sides meeting at \(A\) are \(AB\) and \(AC\).
4. Conclusion
- The three points do form a triangle, and the triangle is a right‑angled triangle (not isosceles or equilateral).
Correct Answer: Yes, the points form a triangle. It is a right‑angled triangle (right angle at (3,2)).