in the figure ab=25m, ad=17m, dp=8m, aq=5m, br=14m and bs=3m. find the area of shaded region
To find the area of the shaded region, we can break it down into separate shapes.
First, let's find the area of triangle ABD.
The formula for finding the area of a triangle is 1/2 * base * height.
The base of triangle ABD is AD, which is 17m.
The height is the perpendicular distance from vertex B to line AD.
To find the height, we can use the Pythagorean theorem.
Using the Pythagorean theorem, we have:
BD^2 = AB^2 - AD^2
BD^2 = 25^2 - 17^2
BD^2 = 625 - 289
BD^2 = 336
BD = √336
BD ≈ 18.33m
Now we can find the height by dropping a perpendicular from B to AD.
Let's call the point where the perpendicular intersects AD as E.
Triangle ABE is a right-angled triangle, and the height of triangle ABD is BE.
Using the Pythagorean theorem for triangle ABE, we have:
AE^2 + BE^2 = AB^2
AE^2 + (18.33)^2 = 25^2
AE^2 + 336.09 ≈ 625
AE^2 ≈ 288.91
AE ≈ √288.91
AE ≈ 16.99m
So, the area of triangle ABD is:
1/2 * base * height
= 1/2 * AD * BE
= 1/2 * 17 * 16.99
≈ 143.65 square meters
Next, let's find the area of quadrilateral BQRS.
This can be divided into two triangles: triangle BQR and triangle SBR.
We can find the area of each triangle separately.
Area of triangle BQR = 1/2 * base * height
= 1/2 * BR * QP
= 1/2 * 14 * 5
= 35 square meters
Area of triangle SBR = 1/2 * base * height
= 1/2 * BR * SB
= 1/2 * 14 * 3
= 21 square meters
Finally, the area of the shaded region is the sum of the areas of triangle ABD, triangle BQR, and triangle SBR.
Area of shaded region = Area of triangle ABD + Area of triangle BQR + Area of triangle SBR
= 143.65 + 35 + 21
= 199.65 square meters
Therefore, the area of the shaded region is approximately 199.65 square meters.
To find the area of the shaded region, we need to determine the areas of the triangles and then subtract them from the area of the rectangle.
First, let's calculate the area of triangle ABD:
The base of triangle ABD is AD, which is 17m, and the height is the perpendicular distance from point B to line AD. From the figure, line BP is perpendicular to AD, so we can use BP as the height.
To find BP, we need to subtract the length of PS from BS:
BP = BS - PS
BP = 3m - 8m (since PS = DP = 8m)
BP = -5m
It's important to note that lengths cannot be negative, so we take the absolute value of BP:
BP = |-5m| = 5m
Now we have the base (AD = 17m) and the height (BP = 5m) of triangle ABD. We can use the formula for the area of a triangle:
Area of triangle = (base * height) / 2
Area of triangle ABD = (17m * 5m) / 2
Area of triangle ABD = 85m²
Next, let's calculate the area of triangle BCS:
The base of triangle BCS is BC, which is equal to the width of the rectangle AB (25m), and the height is BS (3m).
Area of triangle BCS = (BC * BS) / 2
Area of triangle BCS = (25m * 3m) / 2
Area of triangle BCS = 37.5m²
Now we have the areas of both triangles (85m² and 37.5m²). We can add them together to determine the total area of the shaded region:
Total area of the shaded region = Area of triangle ABD + Area of triangle BCS
Total area = 85m² + 37.5m²
Total area = 122.5m²
Therefore, the area of the shaded region is 122.5 square meters.