Distance from Origin to a Line Quiz

Geometry and Measurement

• A parallelogram has sides of 7 and 13, and one of its interior angles is 75⁰; the measurement of its longer diagonal is 15.29.

Triangles

• In triangle DEF, DE = 11 mm, ∠D = 60⁰, and the area is 90 mm²; the measurement of side EF is 18.90.

Quadrilaterals

• The given quadrilateral is a rhombus with interior angles congruent to each other.

Geometric Progression

• In a geometric progression, the fourth term is 1.5, and the eighth term is 24; the 14th term is 768.

Logarithms

• Log P – Log Q is equal to Log P/Q.

Triangle Properties

• In triangle ABC, two sides are 60m and 80m long, and the area is 2,175m²; the third side is 72.10 m.

Time and Clocks

• Ben leaves his office between 1 pm and 2 pm, and after 4-5 hours, he arrives at a time when the hands of the clock are interchanged; he arrives at 1:25.59 pm.

Dilution of Mixture

• A factory produces 95% alcohol per volume; to dilute it to 50% for safe distribution, 5.26 cubic meters of water must be added to 10 cubic meters of mixture.

Ladder and Wall

• A 7m ladder leans against a vertical wall 6m high; the top of the ladder slides downward at 0.5 m/s when the lower end is sliding at 0.5 m/s.

Distances and Lines

• The distance from the origin to the line 8x – 5y – 12 = 0 is 12.83.

Equations and Curves

• The center of the equation x² + y² – 10x – 10y + 25 = 0 is (5, 5).

Number Systems

• The difference of two numbers is 4, and the difference of their squares is equal to 56; the sum of the numbers is 26.

Polygon Angles

• A polygon has a total of 54 diagonals; the sum of its interior angles is 900.

Motion and Velocity

• A car moves at an initial velocity of 3600 m/s and decelerates at 450 m/min²; the distance traveled by the car until it reaches a velocity of 3000 m/s is 264 km.

Integration

• The integral of dx / (5 – 2x) is –(1/2) ln(5 – 2x) + c.

Trigonometry

• The integral of cos(73x) dx with a limit from x = 0 to x = π/6 is 0.148.

Algebraic Equations

• The equation x² = 256 has a solution of x = 4.

Optimization

• The numbers 80 can be divided into two parts so that the product of one part and the cube of the other is at maximum; the parts are x = 20, y = 60.

Cylindrical Tank

• Water flows into a cylindrical tank at a rate of 0.50 m³/sec; the liquid rises at a rate of 0.95 m/s.

Radius of Curvature

• The radius of curvature of the function y = 3x² – 6x + 25 at x = 2 is 3.1.

Lines and Intercepts

• The y-intercept of a line with slope = 4/3 and passing through (-3, 2) is 3.

Area Bounded by Curves

• The area bounded by the curve x² – 4x + 12y – 20 = 0 and the x-axis between x = 1 and x = 2 is 2.11.

Simplification of Expressions

• The integral of e⁷xdx is (1/7)e⁷x + C.

Normal and Tangent Lines

• The equation of the line normal to the curve f(x) = x³ – 1 at (2, 7) is x – 12y + 82 = 0.

Circumference and Area

• The equation of the line tangent to the curve x² + y² = 25 at (4, 3) is 4x + 3y – 25 = 0.