Question

1. The line y = −3x + 5 together with the coordinate axes form a triangle. Determine the area of ​​the triangle. 2. In a right-angled triangle, one of the catheters is 50 cm and the opposite angle is 66∘. Determine the shortest side of the triangle and round to a decimal.

Answer 1

The area of the triangle formed by the line y = -3x + 5 and the coordinate axes is 8.33 square units. The shortest side of the right-angled triangle is 57.14 cm.

Step-by-step explanation:

Determining the Area of the Triangle Formed by the Line and Coordinate Axes:

The line equation is given as y = -3x + 5. To determine the area of the triangle formed by this line and the coordinate axes, we need to find the points where the line intersects the x-axis and the y-axis. These points will form the vertices of the triangle.

To find the x-intercept, set y = 0 and solve for x:

0 = -3x + 5

3x = 5

x = 5/3

So, the line intersects the x-axis at (5/3, 0).

To find the y-intercept, set x = 0 and solve for y:

y = -3(0) + 5

y = 5

So, the line intersects the y-axis at (0, 5).

Now, we have the vertices of the triangle:

A(0, 0) - Origin (vertex shared with both axes)

B(5/3, 0) - x-intercept

C(0, 5) - y-intercept

We can now calculate the area of the triangle formed by these points using the formula for the area of a triangle:

Area = (1/2) * base * height

The base is the distance between points B and C along the x-axis, which is 5/3 units.

The height is the distance between points B and A along the y-axis, which is 5 units.

Area = (1/2) * (5/3) * 5

Area = (5/6) * 5

Area = 25/6 square units

The area of the triangle is 25/6 square units.

Determining the Shortest Side of the Right-Angled Triangle:

In a right-angled triangle, you can use trigonometry to find the shortest side if you know one of the angles and the length of one of the legs.

Given:

One leg (cathetus) = 50 cm

Angle opposite that leg = 66 degrees

The shortest side of the triangle is the side opposite the 66-degree angle, which is the side we want to find.

You can use the trigonometric relationship for a right triangle:

sin(θ) = opposite / hypotenuse

In this case, θ is 66 degrees, the opposite side is the side we want to find (let's call it "x"), and the hypotenuse is the given leg, which is 50 cm.

sin(66°) = x / 50

To find x, multiply both sides by 50:

x = 50 * sin(66°)

Now, calculate the value of x:

x ≈ 50 * 0.9135

x ≈ 45.675

Rounding to one decimal place, the shortest side of the triangle is approximately 45.7 cm.

Learn more about Triangle Calculation