Step 1a

The given arc length is 6 cm and the central angle is 40°

To find the radius of the circle, we can use the formula relating the arc length,

central angle, and radius:

arc length = radius \$times\$ central angle

Explanation: Here we have to find the radius(r) of the circle by using the arc length and central angle
arc length = radius \$times\$ central angle

Step 1b

Substituting these values into the formula, we have: 6 cm = r × 40°

The angle to be in radians rather than°. Since 1 radian is equal to

180° / π, we can convert the angle to radians.

⇒ \$6 "cm" = "r" \times ((40°) times (π / 180°))\$

⇒ \$6 = "r" \times ((2π) / 9) \$ radians

⇒ \$6 / ((2π) / 9) = "r"\$ radians

⇒ \$"r" = (54 "cm") / π\$ radians

⇒ \$"r" = (54 "cm") / 3.14\$ radians

Thus, the radius of the circle is 17.1828 cm.

Explanation: Here, we substitute the given values in the formula. To find the circle’s radius, we substitute the π = 3.14. On solving, we get the radius of the circle is 17.1828cm.

Step 2a

Given the side of a triangle = 12cm

Area of an equilateral triangle = \$\sqrt 3/4\$ a² square units

⇒ \$\sqrt 3/4 \times 12 \times 12\$

⇒ \$\sqrt 3/4 \times 144\$

⇒ \$\sqrt 3 * \cancel144^36/\cancel4^1\$

⇒ \$1.73 \times 36\$

⇒ 62.28 cm²

Explanation: If the side of an equilateral triangle is 12cm, then the area of an equilateral triangle is 62.28 cm².

Step 3a

First, to find the slope (m) of a line passing through two points (x1, y1) and (x2, y2) is given by: \$m = (y2 - y1) / (x2 - x1)\$.

Let’s plug in the coordinates of the given points (1, -2) and (4, -5): \$m = (-5 - (-2)) / (4 - 1)\$ then \$m = (-5 + 2) / 3 \$ ⇒ m = -1.

Explanation: To find the slope (m) of a line passing through two points (x1, y1) and (x2, y2), you can use this formula: m = (y2 - y1) / (x2 - x1)

Using the points (1, -2) and (4, -5), the slope(m) is -1.

Step 3b

In this step, the point-slope form of the equation of a line is:
y − y1 = m(x − x1)

Using the point (1,−2) and the slope m=−1:

⇒y - (-2) = -1(x - 1)

⇒ y + 2 = -1 (x) + 1

⇒ y = -1 (x) + 1 - 2

⇒ y = -1 (x) - 1

Explanation: To proceed, apply the point-slope form. With the slope (m = -1) and one point (1, -2), we can formulate the equation of the line as follows: y − y1 = m(x − x1). Therefore, substituting the values, we get y = -1x - 1.

Step 3c

So, the equation of the line passing through the points (1, -2) and (4, -5) is y = -x - 1.

Explanation: The equation for the line that passes through the coordinates (1, -2) and (4, -5) is y = -x - 1.