One side of rectangular field is 15 meters and one of its diagonals is 17 meters. Then find the area of the field.
120 m 2
Finding the Area of a Rectangular Field
The question asks us to find the area of a rectangular field given the length of one side and the length of its diagonal. A rectangle has four sides, with opposite sides being equal in length, and all interior angles are 90 degrees. The diagonal of a rectangle divides it into two right-angled triangles.
Using the Pythagorean Theorem for Rectangle Dimensions
Let the rectangular field have a length $l$ and a width $w$. We are given that one side is 15 meters. Let's assume $l = 15$ m. The diagonal ($d$) is given as 17 meters.
In a rectangle, the diagonal, one length side, and one width side form a right-angled triangle. The diagonal is the hypotenuse of this triangle, and the length and width are the other two sides (legs).
According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In our case, this translates to:
\(l^2 + w^2 = d^2\)
We know $l = 15$ m and $d = 17$ m. We need to find the value of $w$.
\(15^2 + w^2 = 17^2\)
Calculate the squares:
\(225 + w^2 = 289\)
Now, isolate $w^2$ by subtracting 225 from both sides:
\(w^2 = 289 - 225\)
\(w^2 = 64\)
To find $w$, take the square root of both sides:
\(w = \sqrt{64}\)
\(w = 8\)
So, the width of the rectangular field is 8 meters.
Calculating the Area of the Rectangular Field
The area of a rectangle is calculated by multiplying its length and width:
\(\text{Area} = l \times w\)
We have $l = 15$ m and $w = 8$ m.
\(\text{Area} = 15 \, \text{m} \times 8 \, \text{m}\)
\(\text{Area} = 120 \, \text{m}^2\)
The area of the rectangular field is 120 square meters.
Summary of Steps to Find Rectangle Area
- Identify the given information: one side (length or width) and the diagonal.
- Recognize the formation of a right-angled triangle using the sides and the diagonal.
- Apply the Pythagorean theorem ($a^2 + b^2 = c^2$) to find the missing side.
- Use the formula for the area of a rectangle (Area = length $\times$ width) with the calculated dimensions.
| Property | Value | Unit |
|---|---|---|
| Given Side (Length) | 15 | meters |
| Diagonal | 17 | meters |
| Calculated Side (Width) | 8 | meters |
| Area | 120 | m2 |
Revision Table: Key Concepts for Rectangle Problems
| Concept | Formula | Description |
|---|---|---|
| Area of Rectangle | \(A = l \times w\) | Product of length and width. |
| Perimeter of Rectangle | \(P = 2(l + w)\) | Sum of lengths of all four sides. |
| Diagonal of Rectangle | \(d = \sqrt{l^2 + w^2}\) | Length of the line segment connecting opposite vertices (using Pythagorean theorem). |
Additional Information: Applications of Pythagorean Theorem
The Pythagorean theorem is a fundamental concept in geometry and is not limited to finding dimensions of rectangles. It applies to any right-angled triangle. It is used in various fields:
- Construction: Ensuring corners are square (90 degrees).
- Navigation: Calculating distances.
- Engineering: Designing structures and solving spatial problems.
- Computer Graphics: Calculating distances and positions.
Understanding the relationship between the sides of a right triangle is crucial for many geometric and real-world problems.
Important Questions from Plane Figures
If the area of a square is 625 cm 2, then what is the perimeter of the square?
The area and the perimeter of a sheet of paper are 240 cm 2and 68 cm, respectively. What would be its length and breadth?
The bisector of ∠B in ΔABC meets AC at D. If AB = 12 cm, BC = 18 cm and AC = 15 cm, then the length of AD (in cm) is:
The perimeter and the length of one of the diagonals of a rhombus is 26 cm and 5 cm respectively. Find the length of its other diagonal (in cm).
ABCD is a trapezium in which AB is parallel to DC. Let E and F be the midpoints on AD and BC respectively. If EF = 10 cm and AB - DC = 4 cm, then what is the value of AB × DC?
