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
3621
What is the perpendicular distance from the origin (0,0) to the line 3x + 4y - 10 = 0?
Answer:
2
Step 1: The distance formula from a point (x1, y1) to a line ax + by + c = 0 is d = |ax1 + by1 + c| / √(a² + b²). Step 2: Substitute (0,0) into the line equation: d = |3(0) + 4(0) - 10| / √(3² + 4²). Step 3: Simplify: d = |-10| / √25 = 10 / 5 = 2.
3622
The centroid of a triangle represents the intersection point of its:
Answer:
Medians
Step 1: A median is a line segment drawn from a vertex to the midpoint of the opposite side. Step 2: Every triangle has three medians. Step 3: The single point where all three medians intersect is defined geometrically as the centroid.
3623
The incenter of a triangle represents the intersection point of its:
Answer:
Angle bisectors
Step 1: Different centers of a triangle are formed by different intersecting lines. Step 2: The centroid is from medians, orthocenter from altitudes, and circumcenter from perpendicular bisectors. Step 3: The incenter, the center of the inscribed circle, is the intersection of the angle bisectors.
3624
If the centroid of a triangle is at the origin (0,0) and two of its vertices are (2, 3) and (-1, -1), what is the third vertex?
Answer:
(-1, -2)
Step 1: Set up the centroid equations. Let the third vertex be (x,y). (2 - 1 + x)/3 = 0 and (3 - 1 + y)/3 = 0. Step 2: Solve for x: (1 + x)/3 = 0 => 1 + x = 0 => x = -1. Step 3: Solve for y: (2 + y)/3 = 0 => 2 + y = 0 => y = -2. The vertex is (-1, -2).
3625
Find the centroid of a triangle with vertices at (a, 0), (0, b), and the origin.
Answer:
(a/3, b/3)
Step 1: Identify the three vertices: (a, 0), (0, b), and (0, 0). Step 2: Apply the centroid formula: x = (a + 0 + 0) / 3 = a/3. Step 3: Apply for y: y = (0 + b + 0) / 3 = b/3. The centroid is (a/3, b/3).
3626
The centroid of a triangle divides each median in the ratio of:
Answer:
2:1
Step 1: The centroid is the point where the three medians of a triangle intersect. Step 2: A core geometric theorem states that the centroid acts as a balance point on the medians. Step 3: It divides each median into two segments in a 2:1 ratio, with the larger segment connecting to the vertex.
3627
What is the centroid of the triangle with vertices (1, 2), (3, -4), and (5, 8)?
Answer:
(3, 2)
Step 1: Use the centroid formula C = ((x1+x2+x3)/3, (y1+y2+y3)/3). Step 2: Sum the x-coordinates: (1 + 3 + 5)/3 = 9/3 = 3. Step 3: Sum the y-coordinates: (2 + (-4) + 8)/3 = 6/3 = 2. The centroid is (3, 2).
3628
Find the centroid of the triangle with vertices (0, 0), (3, 0), and (0, 3).
Answer:
(1, 1)
Step 1: The centroid formula is ((x1+x2+x3)/3, (y1+y2+y3)/3). Step 2: Calculate x: (0 + 3 + 0) / 3 = 3 / 3 = 1. Step 3: Calculate y: (0 + 0 + 3) / 3 = 3 / 3 = 1. The centroid is (1, 1).
3629
What is the reflection of the point (2, 3) across the x-axis?
Answer:
(2, -3)
Step 1: Reflecting a point across the x-axis means the horizontal position stays the same, but it flips vertically. Step 2: Therefore, the x-coordinate remains unchanged, and the y-coordinate reverses its sign. Step 3: The point (2, 3) becomes (2, -3).
3630
Find the point dividing the segment between (-1, 2) and (3, -2) internally in the ratio 1:3.
Answer:
(0, 1)
Step 1: Apply the section formula with m=1, n=3. Step 2: x = (1*3 + 3*(-1)) / (1+3) = (3 - 3) / 4 = 0. Step 3: y = (1*(-2) + 3*2) / (1+3) = (-2 + 6) / 4 = 4 / 4 = 1. The point is (0, 1).