Question 4
🖼️ Visualizing the Triangle
Figure 1: The points A, B, and C plotted on the plane.
📋 Given Information
- First vertex \( A = (5, -2) \)
- Second vertex \( B = (6, 4) \)
- Third vertex \( C = (7, -2) \)
💡 Condition for Isosceles Triangle
A triangle is isosceles if at least two of its sides are equal in length.
📝 Step-by-Step Solution
Using Distance Formula for \( A(5, -2) \) and \( B(6, 4) \):
\[ AB = \sqrt{(6 – 5)^2 + (4 – (-2))^2} \] \[ AB = \sqrt{(1)^2 + (6)^2} \] \[ AB = \sqrt{1 + 36} = \sqrt{37} \]
Using Distance Formula for \( B(6, 4) \) and \( C(7, -2) \):
\[ BC = \sqrt{(7 – 6)^2 + (-2 – 4)^2} \] \[ BC = \sqrt{(1)^2 + (-6)^2} \] \[ BC = \sqrt{1 + 36} = \sqrt{37} \]Using Distance Formula for \( C(7, -2) \) and \( A(5, -2) \):
\[ CA = \sqrt{(5 – 7)^2 + (-2 – (-2))^2} \] \[ CA = \sqrt{(-2)^2 + (0)^2} \] \[ CA = \sqrt{4} = 2 \]🔧 Side Length Comparison
Figure 2: Since sides AB and BC are equal (√37), the triangle is isosceles.
From the calculations:
\[ AB = \sqrt{37} \] \[ BC = \sqrt{37} \] \[ CA = 2 \]Since \( AB = BC \), the triangle has two equal sides.
❓ Frequently Asked Questions (FAQ)
An isosceles triangle is a triangle that has at least two sides of equal length. The angles opposite these equal sides are also equal.
Yes, an equilateral triangle (all three sides equal) is a special case of an isosceles triangle because it satisfies the condition of having “at least two” equal sides.
For proving it is isosceles, finding two equal sides is enough. However, finding the third side helps confirm it isn’t an equilateral triangle (a specific type of isosceles).