Circumference = 2πr => r = circumference / (2π) = 31.4 / (2 * 3.14) = 5 cm

Using Heron's formula, s = (5 + 12 + 13)/2 = 15. Area = √[s(s-a)(s-b)(s-c)] = √[15(15-5)(15-12)(15-13)] = √[15*10*3*2] = √[900] = 30 cm².

Using the Pythagorean theorem, 7^2 + 24^2 = 49 + 576 = 625 = 25^2. Therefore, triangle ABC is a right triangle.

This is a right triangle. Area = 1/2 * base * height = 1/2 * 5 * 12 = 30 cm².

The distance between the centers of two externally tangent circles is the sum of their radii: 3 + 4 = 7 cm.

Distance between centers = r1 + r2 = 3 cm + 5 cm = 8 cm.

The distance between the centers of two externally tangent circles is the sum of their radii: 3 + 4 = 7 cm.

If two lines are perpendicular, the sum of their angles is 90 degrees. Therefore, if one line makes an angle of 30 degrees with the horizontal, the other line must make an angle of 90 - 30 = 60 degrees with the horizontal.

Perpendicular lines intersect at right angles, which measure 90 degrees.

Vertically opposite angles are equal. Therefore, the measure of the vertically opposite angle is also 70 degrees.

Adjacent angles formed by intersecting lines are supplementary. Therefore, 180 - 70 = 110 degrees.

Alternate interior angles are equal when two parallel lines are cut by a transversal. Therefore, if one angle is 65 degrees, the other alternate interior angle is also 65 degrees.

Area = πr² = 3.14 * (4)² = 3.14 * 16 = 50.24 cm²

Area of sector = (θ/360) × πr² = (90/360) × π × 10² = (1/4) × 100π = 25π cm².

Area of sector = (θ/360) × πr² = (90/360) × π(6)² = (1/4) × 36π = 9π cm².

Area = side² = 8² = 64 cm²

Area = (√3/4) * side² = (√3/4) * 6² = (√3/4) * 36 = 9√3 cm².

Circumference = 2 * π * radius = 2 * π * 7 = 14π cm

The standard form of a circle's equation is (x-h)² + (y-k)² = r². Here, h=2, k=-3, r=4. Thus, (x-2)² + (y+3)² = 16.

The standard form of a circle's equation is (x-h)² + (y-k)² = r². Here, h=3, k=-2, r=5, so (x-3)² + (y+2)² = 25.

Arc length = (θ/360) × 2πr = (60/360) × 2π(5) = (1/6) × 10π = 5π/6 cm.

In a right triangle, the altitude from the right angle to the hypotenuse is equal to the length of the other side. Thus, the altitude is 6 cm.

Diagonal = √(length² + width²) = √(3² + 4²) = √(9 + 16) = √25 = 5 cm

Diagonal = √(length² + width²) = √(6² + 8²) = √(36 + 64) = √100 = 10 cm

Area = πr². Given area = 50π, r² = 50, r = √50 = 5√2. Diameter = 2r = 10√2 units.

Using the median formula: m_a = 1/2 * sqrt(2b^2 + 2c^2 - a^2), where a = BC, b = AC, c = AB. m_a = 1/2 * sqrt(2*6^2 + 2*10^2 - 8^2) = 7 cm.

Supplementary angles sum up to 180 degrees. Therefore, the angle measures 180 - 110 = 70 degrees.

Complementary angles sum up to 90 degrees. If one angle is 30 degrees, the other angle is 90 - 30 = 60 degrees.

The measure of each interior angle in a regular hexagon can be calculated using the formula (n-2) * 180/n, where n is the number of sides. For a hexagon, n=6, so the measure is (6-2) * 180/6 = 120 degrees.

A regular quadrilateral is a square, and each angle in a square measures 90 degrees.