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 length of a rectangle is increased to its three times and breadth is decreased to its half, then the ratio of the area of given rectangle to the area of new rectangle is:
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What is the area of the square (in cm 2) whose vertices lie on a circle of radius 5 cm?
The circumcentre of an equilateral triangle is at a distance of 3.2 cm from the base of the triangle. What is the length (in cm) of each of its altitudes?
The perimeter of a circular lawn is 1232 m. There is 7 m wide path around the lawn. The area (in m 2) of the path is:
Take \(\left(\pi=\frac{22}{7}\right)\)
