Final answer:
The value of t is 2, which is the y-coordinate of vertex C of the right angled triangle formed by vertices A (5,2), B (2,-2), and C (-2,t), with B being the right angle.
Step-by-step explanation:
To find the value of t for vertex C (-2, t) in a right angled triangle with vertices A (5, 2), B (2, -2), and the given condition that ∠ B = 90 degrees, we can use the distance formula to calculate the length of the sides AB, BC, and AC. Since B is the right angle, AB and BC will be the perpendicular sides, and AC will be the hypotenuse.
First, we find the lengths of AB and BC:
- AB = √[(5 - 2)² + (2 - -2)²] = √[3² + 4²] = √[9 + 16] = √25 = 5.
- BC is reliant on the value of t, so BC = √[(2 - -2)² + (-2 - t)²] = √[4² + (-2 - t)²] = √[16 + (-2 - t)²].
- AC = √[(5 - -2)² + (2 - t)²] = √[7² + (2 - t)²] = √[49 + (2 - t)²].
Applying Pythagoras' theorem to these lengths (since AB² + BC² = AC² for a right angled triangle), and solving for t, we get the following equation:
5² + (√[16 + (-2 - t)²])² = (√[49 + (2 - t)²])²
25 + (16 + (-2 - t)²) = (49 + (2 - t)²).
Simplifying the equation:
41 + (-2 - t)² = 49 + (2 - t)².
After solving for t, we get t = 2 as t must be positive because t lies on the y-axis above point B(-2,-2).