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
3691
Find the coordinates of the point that divides the line segment joining (1, 2) and (3, 6) internally in the ratio 1:1.
Answer:
(2, 4)
Step 1: Dividing a segment in a 1:1 ratio is the same as finding the midpoint. Step 2: x = (1 + 3)/2 = 4/2 = 2. Step 3: y = (2 + 6)/2 = 8/2 = 4. The point is (2, 4).
3692
What is the midpoint of the line segment connecting (-5, 7) and (3, -1)?
Answer:
(-1, 3)
Step 1: Use the midpoint formula. Step 2: x = (-5 + 3)/2 = -2/2 = -1. Step 3: y = (7 + (-1))/2 = 6/2 = 3. The midpoint is (-1, 3).
3693
One endpoint of a circle's diameter is (1, 2) and its center is (3, 4). What is the other endpoint?
Answer:
(5, 6)
Step 1: Let the other endpoint be (x, y). The center is the midpoint of the diameter. Step 2: Set up equations: (1 + x)/2 = 3 and (2 + y)/2 = 4. Step 3: Solve for x and y: 1 + x = 6 => x = 5; 2 + y = 8 => y = 6. The point is (5, 6).
3694
What is the midpoint of the line segment joining (a, b) and (a, -b)?
Answer:
(a, 0)
Step 1: Use the midpoint formula: M = ((x1 + x2)/2, (y1 + y2)/2). Step 2: Calculate x: (a + a)/2 = 2a/2 = a. Step 3: Calculate y: (b + (-b))/2 = 0/2 = 0. The midpoint is (a, 0).
3695
Find the midpoint of the line segment connecting (-3, -3) and (3, 3).
Answer:
(0, 0)
Step 1: Apply the midpoint formula. Step 2: x-coordinate = (-3 + 3)/2 = 0/2 = 0. Step 3: y-coordinate = (-3 + 3)/2 = 0/2 = 0. The midpoint is the origin (0, 0).
3696
What is the midpoint of the line segment joining (2, 4) and (6, 8)?
Answer:
(4, 6)
Step 1: Use the midpoint formula: M = ((x1 + x2)/2, (y1 + y2)/2). Step 2: Substitute the given coordinates: x = (2 + 6)/2 = 8/2 = 4; y = (4 + 8)/2 = 12/2 = 6. Step 3: The midpoint is (4, 6).
3697
What is the perimeter of the triangle with vertices (0, 0), (3, 0), and (0, 4)?
Answer:
12
Step 1: Calculate the lengths of the three sides using the distance formula. Step 2: Side 1 (0,0 to 3,0) = 3. Side 2 (0,0 to 0,4) = 4. Side 3 (3,0 to 0,4) = √[3² + 4²] = 5. Step 3: Perimeter = sum of sides = 3 + 4 + 5 = 12.
3698
Which point on the x-axis is equidistant from the points (2, -5) and (-2, 9)?
Answer:
(-7, 0)
Step 1: Let the point on the x-axis be P(x, 0). Step 2: Set distances equal: √[(x - 2)² + (0 + 5)²] = √[(x + 2)² + (0 - 9)²]. Step 3: Square both sides: x² - 4x + 4 + 25 = x² + 4x + 4 + 81. Step 4: Simplify to -8x = 56, yielding x = -7. Point is (-7, 0).
3699
What is the distance between the points (-3, 4) and (3, -4)?
Answer:
10
Step 1: Use the distance formula d = √[(x2 - x1)² + (y2 - y1)²]. Step 2: Substitute coordinates: d = √[(3 - (-3))² + (-4 - 4)²]. Step 3: Simplify: d = √[6² + (-8)²] = √[36 + 64] = √100 = 10.
3700
Find the distance between the points (1, 1) and (13, 6).
Answer:
13
Step 1: Use the distance formula. Step 2: d = √[(13 - 1)² + (6 - 1)²]. Step 3: Simplify to d = √[12² + 5²] = √[144 + 25] = √169 = 13.