Find the area of the triangle whose height is 56 cm and the corresponding base is 1/4th of the height.
392 cm2
Calculating the Area of a Triangle
The question asks us to find the area of a triangle given its height and the relationship of its base to the height.
Understanding the Given Information
- The height of the triangle is 56 cm.
- The corresponding base of the triangle is $\frac{1}{4}\text{th}$ of the height.
Step 1: Calculate the Length of the Base
The base is given as $\frac{1}{4}$ of the height. We can calculate the base by multiplying the height by $\frac{1}{4}$.
Base = $\frac{1}{4} \times \text{Height}$
Base = $\frac{1}{4} \times 56 \text{ cm}$
To calculate this, we can divide 56 by 4.
$$ \text{Base} = \frac{56}{4} \text{ cm} $$ $$ \text{Base} = 14 \text{ cm} $$So, the base of the triangle is 14 cm.
Step 2: Apply the Formula for the Area of a Triangle
The formula for the area of a triangle is:
$$ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} $$Now, we substitute the values we know:
- Base = 14 cm
- Height = 56 cm
We can simplify the calculation:
$$ \text{Area} = ( \frac{1}{2} \times 14 ) \times 56 \text{ cm}^2 $$ $$ \text{Area} = 7 \times 56 \text{ cm}^2 $$Now, we multiply 7 by 56:
$$ 7 \times 56 = 7 \times (50 + 6) $$ $$ = (7 \times 50) + (7 \times 6) $$ $$ = 350 + 42 $$ $$ = 392 $$So, the area of the triangle is 392 cm$^2$.
Final Answer
The area of the triangle with a height of 56 cm and a base $\frac{1}{4}\text{th}$ of the height is 392 cm$^2$.
| Measurement | Value |
|---|---|
| Height | 56 cm |
| Base (calculated) | 14 cm |
| Area (calculated) | 392 cm$^2$ |
Revision Table: Key Triangle Concepts
| Concept | Description | Formula (if applicable) |
|---|---|---|
| Triangle | A polygon with three sides and three angles. | - |
| Base | Any side of the triangle chosen as the base for area calculation. | - |
| Height (Altitude) | The perpendicular distance from the opposite vertex to the base (or the extension of the base). | - |
| Area of Triangle | The measure of the space enclosed by the triangle. | $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ |
Additional Information: Types of Triangles
Triangles can be classified based on their sides or angles.
Classification by Sides:
- Equilateral Triangle: All three sides are equal in length. All angles are 60 degrees.
- Isosceles Triangle: Two sides are equal in length. The angles opposite the equal sides are also equal.
- Scalene Triangle: All three sides have different lengths. All three angles have different measures.
Classification by Angles:
- Acute-angled Triangle: All three angles are acute (less than 90 degrees).
- Right-angled Triangle: One angle is a right angle (exactly 90 degrees). The side opposite the right angle is called the hypotenuse.
- Obtuse-angled Triangle: One angle is obtuse (greater than 90 degrees).
The formula for the area ($\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$) works for all types of triangles, as long as the correct corresponding base and height are used.
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?
One side of rectangular field is 15 meters and one of its diagonals is 17 meters. Then find the area of the field.
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).
