Find the area of the shaded region.
diagram is a rectangle with 10 height and 17 base. there is a triangle inscribed that is not shaded.
its a little confusing and im supposed to assume 30-60 -90 triangles??
To find the area of the shaded region in the given diagram, we need to determine the area of the rectangle and subtract the area of the triangle.
First, let's find the area of the rectangle. The base of the rectangle is 17 units and the height is 10 units. Therefore, the area of the rectangle is given by:
Area of rectangle = base × height = 17 × 10 = 170 square units.
Next, let's determine the area of the triangle. Since it is mentioned that we can assume the presence of a 30-60-90 triangle, we can use that information to find the dimensions of the triangle.
In a 30-60-90 triangle, the length of the hypotenuse (the longest side) is twice the length of the shorter leg (opposite the 30-degree angle). Similarly, the length of the longer leg (opposite the 60-degree angle) is √3 times the length of the shorter leg.
Since the triangle is inscribed within the rectangle, one of the shorter legs of the triangle matches one side of the rectangle, which has a length of 10 units. Therefore, the length of the hypotenuse is 2 × 10 = 20 units, and the length of the longer leg is √3 × 10 = 10√3 units.
Now, we can use the dimensions of the triangle to determine its area. The formula to calculate the area of a triangle is ½ × base × height. In this case, the base of the triangle is 10 units, and the height is 10√3 units.
Area of triangle = ½ × base × height = ½ × 10 × 10√3 = 50√3 square units.
Finally, to find the area of the shaded region, we subtract the area of the triangle from the area of the rectangle:
Area of shaded region = Area of rectangle - Area of triangle
= 170 - 50√3
≈ 170 - 86.60
≈ 83.40 square units.
Therefore, the area of the shaded region in the given diagram is approximately 83.40 square units.
To find the area of the shaded region, we first need to find the area of the rectangle and the area of the triangle.
The area of a rectangle is calculated by multiplying its base by its height. In this case, the base is 17 and the height is 10. So the area of the rectangle is 17 * 10 = 170 square units.
To calculate the area of the triangle, we need to determine the length of the triangle's sides.
Since we are assuming the triangle is a 30-60-90 triangle, the ratio of the sides is as follows:
- The shorter leg is opposite the 30-degree angle and has a length of x.
- The longer leg is opposite the 60-degree angle and has a length of x * √3.
- The hypotenuse is opposite the 90-degree angle and has a length of 2x.
In this case, the base of the rectangle is the hypotenuse of the triangle, so its length is 2x = 17.
Solving for x, we get x = 17 / 2 = 8.5.
Now we can determine the length of the shorter leg of the triangle, which is equal to half the base of the rectangle: x = 8.5.
The area of a triangle is calculated by multiplying the length of its base by its height and then dividing by 2. In this case, the base of the triangle is 17 (which is the same as the base of the rectangle), and the height is 8.5. So the area of the triangle is (17 * 8.5) / 2 = 72.25 square units.
Finally, we subtract the area of the triangle from the area of the rectangle to find the area of the shaded region: 170 - 72.25 = 97.75 square units.
Therefore, the area of the shaded region is 97.75 square units.