Geometry word problems

Solution:

• Let x be the measure of the first angle
• Then, the second angle is 2x
• Since the angles are supplementary, they add up to 180°
• x + 2x = 180°
• 3x = 180°
• Since 3 × 60 = 180, x = 60
• The measure of the first angle is 60°
• The measure of the second is 2x = 2 × 60 = 120°

Solution:

• If the x-coordinate or the y-coordinate is the same for all points, then the points are collinear.
• After a close inspection, we see that the x-coordinate is the same for all points (2).
• If all points have the same x-coordinate or y-coordinate, they are colinear. Therefore, the points are collinear.

Solution:

• If the perimeter is 8 cm, then the length of one side is 2 cm since 2 cm + 2 cm + 2 cm + 2 cm = 8 cm
• Area = 2 cm × 2 cm = 4 cm²

Solution:

• Thing to know: The sum of the angles in a triangle is equal to 180°
• Meaning of the ratio 1:3: The second acute angle is 3 times bigger than the first acute angle
• Let x be the first acute angle, then the second acute angle will be 3x
• x + 3x + 90° = 180° (right angle is 90°)
• 4x + 90° = 180°
• 4x = 90°
• x = 22.5° (since 4 × 22.5 = 90°)
• The second angle is 3x = 3 × 22.5° = 67.5°
• The measures of the two acute angles are 22.5° and 67.5°

Solution:

• Let's find the x-coordinate first:
• (x₁ + 4) / 2 = 3
• x₁ = 2 since 2 + 4 = 6 and 6 ÷ 2 = 3
• Now for the y-coordinate:
• (y₁ + 7) / 2 = 6
• y₁ = 5 since 5 + 7 = 12 and 12 ÷ 2 = 6
• Therefore, the other endpoint is (2, 5)

Solution:

• Formula: Sum of angles in an n-gon = (n - 2) × 180°
• 2340° = (n - 2) × 180°
• 2340° = 180°n - 360°
• 2340° + 360° = 180°n
• 2700° = 180°n
• 2700° ÷ 180° = n
• 15 = n
• Therefore, the n-gon has 15 sides

Solution:

• When two lines are perpendicular, their slopes are negative reciprocals:
• m₁ × m₂ = -1 (where m₁ and m₁ are the slopes)
• m₁ × 5 = -1
• Divide both sides by 5:
• m₁ = -1 ÷ 5 = -0.20
• Therefore, the slope of the first line is -0.20

Solution:

• We'll use the formula A = πr² for circular area
• First, let's identify the radii:
  • Penny: r = 0.375 inch (half of 0.750)
  • Quarter: r = 0.4775 inch (half of 0.955)
• Let B be the uncovered area:
• B = area of quarter - area of penny
• B = π(0.4775)² - π(0.375)²
• B = 0.715 - 0.441
• B = 0.274 square inches
• Note: As long as the penny remains entirely within the quarter's circumference, the uncovered area remains constant regardless of the penny's position.