 Find the area of a quadrilateral whose sides are 9m, 40m, 28m and 15m respectively and the angles between first two sides is a right angle.

To find the area of a quadrilateral with sides 9m, 40m, 28m, and 15m, and given that the angle between the first two sides (9m and 40m) is a right angle, we can break this problem into two parts. We treat the quadrilateral as two triangles and compute their areas.

### Step 1: Divide the quadrilateral into two triangles
- One triangle has sides 9m and 40m, with a right angle between them.
- The second triangle has sides 28m and 15m, but we need to calculate the length of the diagonal (hypotenuse) from the first triangle.

#### Step 2: Calculate the diagonal using Pythagoras' theorem
For the right-angled triangle formed by sides 9m and 40m:
\[
d = \sqrt{9^2 + 40^2} = \sqrt{81 + 1600} = \sqrt{1681} = 41 \text{m}
\]
So, the diagonal of the quadrilateral is 41m.

### Step 3: Use Heron’s formula to find the area of the second triangle
Now, we consider the second triangle, which has sides 28m, 15m, and 41m.

First, calculate the semi-perimeter \( s \) of the second triangle:
\[
s = \frac{28 + 15 + 41}{2} = \frac{84}{2} = 42 \text{m}
\]
Now, apply Heron's formula for the area of the second triangle:
\[
A = \sqrt{s(s - 28)(s - 15)(s - 41)}
\]
\[
A = \sqrt{42(42 - 28)(42 - 15)(42 - 41)} = \sqrt{42 \times 14 \times 27 \times 1} = \sqrt{15876} \approx 126 \text{m}^2
\]

### Step 4: Calculate the area of the first triangle
For the first triangle, the area is simpler to calculate because it is a right-angled triangle:
\[
A = \frac{1}{2} \times 9 \times 40 = 180 \text{m}^2
\]

### Step 5: Total area of the quadrilateral
Now, sum the areas of the two triangles:
\[
\text{Total Area} = 180 \text{m}^2 + 126 \text{m}^2 = 306 \text{m}^2
\]

Thus, the area of the quadrilateral is **306 square meters**.

Would you like more details on any part of the solution?

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Here are five related questions you might find interesting:

1. How do you apply Heron’s formula to any general triangle?
2. What is the significance of the Pythagorean theorem in solving geometry problems?
3. Can you find the area of a triangle if only the angles are given?
4. How do you calculate the area of a quadrilateral with non-right angles?
5. How does the diagonal affect the properties of a quadrilateral?

**Tip**: Always check if a given shape can be simplified into familiar forms (like triangles) to make calculations easier.

Find the area of a quadrilateral whose sides are 9m, 40m, 28m and 15m respectively and the angles between the first two sides is a right angle.

Learn how to calculate the area of a quadrilateral with sides 9m, 40m, 28m, and 15m, using a right angle between two sides and Heron's formula. This detailed step-by-step guide covers geometry, including the Pythagorean theorem and Heron’s formula. Suitable for students in Grades 9-10.

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