All Categories MCQs
Topic Notes: All Categories
General Description
Plato
- Biography: Ancient Greek philosopher (427–347 BCE), student of Socrates and teacher of Aristotle, founder of the Academy in Athens.
- Important Ideas:
- Theory of Forms
- Philosopher-King
- Ideal State
3641
What is the area of a triangle with vertices (0, 0), (a, 0), and (0, b), where a, b > 0?
Answer:
ab / 2
Step 1: The given points form a right-angled triangle at the origin. Step 2: The base along the x-axis has length 'a' and the height along the y-axis has length 'b'. Step 3: Using the area formula (1/2 * base * height), the area is ab / 2.
3642
If the points (0, 0), (1, 1), and (2, x) are collinear, what is the value of x?
Answer:
2
Step 1: For points to be collinear, the slope between the first two must equal the slope between the second and third. Step 2: Slope between (0,0) and (1,1) is (1-0)/(1-0) = 1. Step 3: Slope between (1,1) and (2,x) is (x-1)/(2-1) = x-1. Set x-1 = 1, giving x = 2.
3643
What is the area of a polygon with vertices at (0, 0), (2, 0), (2, 2), and (0, 2)?
Answer:
4
Step 1: Identify the shape formed by the coordinates. The points form a square with side lengths of 2 units. Step 2: The formula for the area of a square or rectangle is base * height. Step 3: Area = 2 * 2 = 4.
3644
The standard formula for the area of a triangle given its three vertex coordinates uses concepts from which mathematical structure?
Answer:
Determinant of a matrix
Step 1: The formula for the area of a triangle with vertices (x1,y1), (x2,y2), (x3,y3) is 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|. Step 2: This exact algebraic structure is derived from evaluating a 3x3 determinant composed of the points' coordinates and a column of ones. Step 3: Thus, it uses the determinant of a matrix.
3645
Find the area of the triangle with vertices (-3, 0), (3, 0), and (0, 4).
Answer:
12
Step 1: The base of the triangle lies on the x-axis from x = -3 to x = 3. Length of base = 3 - (-3) = 6. Step 2: The third vertex is at (0,4), so the height (perpendicular distance to x-axis) is 4. Step 3: Area = (1/2) * base * height = (1/2) * 6 * 4 = 12.
3646
If the calculated area of a triangle using coordinate geometry formulas is exactly zero, what can be concluded about the three points?
Answer:
They are collinear
Step 1: The area of a triangle represents the space enclosed by three non-collinear points. Step 2: If the area is exactly zero, it means no space is enclosed. Step 3: This implies that the three points lie on a single straight line, meaning they are collinear.
3647
What is the area of a triangle with vertices (1, 1), (1, 4), and (5, 1)?
Answer:
6
Step 1: This is a right triangle since two points share the same x-coordinate (1) forming a vertical segment, and two share the same y-coordinate (1) forming a horizontal segment. Step 2: Height = |4 - 1| = 3. Base = |5 - 1| = 4. Step 3: Area = (1/2) * 4 * 3 = 6.
3648
Calculate the area of the triangle whose vertices are (0, 0), (5, 0), and (0, 12).
Answer:
30
Step 1: The vertices form a right-angled triangle positioned at the origin. Step 2: The base lies along the x-axis with length 5, and the height lies along the y-axis with length 12. Step 3: Area = (1/2) * base * height = (1/2) * 5 * 12 = 30.
3649
What is the x-intercept of a line that is parallel to the x-axis?
Answer:
No x-intercept
Step 1: A line parallel to the x-axis is perfectly horizontal. Step 2: Unless it is the x-axis itself, it will never cross or intersect the x-axis. Step 3: Thus, a parallel line has no x-intercept.
3650
What is the length of the portion of the line 3x + 4y = 12 intercepted between the coordinate axes?
Answer:
5
Step 1: Find the intercepts. x-intercept (y=0) is 3x=12 => x=4. y-intercept (x=0) is 4y=12 => y=3. The points are (4,0) and (0,3). Step 2: Use the distance formula to find the length between these two points. Step 3: Length = √[(4-0)² + (0-3)²] = √[16 + 9] = √25 = 5.