Test Questions for Verifying Isosceles and Right Triangles with Coordinates.

📚 Quick Study Guide

• 📏Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• 📐Isosceles Triangle: A triangle with at least two sides of equal length. Calculate the lengths of all three sides using the distance formula; if two are equal, it's isosceles.
• ➕Right Triangle: A triangle containing one 90-degree angle. Use the Pythagorean theorem to check if $a^2 + b^2 = c^2$, where $a$ and $b$ are the lengths of the shorter sides, and $c$ is the length of the longest side (hypotenuse).
• 🧭Slope Formula: The slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $m = \frac{y_2 - y_1}{x_2 - x_1}$.
• ❗Perpendicular Lines: Two lines are perpendicular if the product of their slopes is -1 (i.e., $m_1 * m_2 = -1$). This can also be used to determine a right angle in a triangle.

Practice Quiz

1. What coordinates define a triangle that is both isosceles and right?
   1. (0,0), (5,0), (0,5)
   2. (1,1), (4,1), (1,3)
   3. (2,2), (5,2), (2,4)
   4. (3,3), (6,3), (3,5)
2. Given points A(1, 2), B(4, 6), and C(7, 2), what type of triangle is ABC?
   1. Equilateral
   2. Isosceles
   3. Right
   4. Scalene
3. Which set of points forms a right triangle?
   1. (0, 0), (1, 1), (2, 2)
   2. (0, 0), (1, 0), (0, 2)
   3. (1, 1), (2, 3), (4, 5)
   4. (2, 2), (3, 4), (5, 6)
4. If a triangle has vertices at (1,1), (5,1) and (3,4), is it isosceles?
   1. Yes
   2. No
   3. Cannot be determined
   4. Only if it is also a right triangle
5. Triangle PQR has vertices P(0,0), Q(4,0) and R(2,2\sqrt{3}). Is it equilateral, isosceles, or right angled?
   1. Equilateral
   2. Isosceles
   3. Right Angled
   4. None of the above
6. What are the conditions for a triangle to be classified as both isosceles and right angled using coordinates?
   1. Two sides must be equal and two angles must be 45 degrees.
   2. Two sides must be equal and follow Pythagorean theorem.
   3. Two angles must be 45 degrees and follow Pythagorean theorem.
   4. All of the above.
7. Given coordinates (1, 1), (4, 5), and (8, 2), is the resulting triangle a right triangle?
   1. Yes
   2. No
   3. Cannot be determined
   4. Only if it's also Isosceles