Final answer:
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