Triangle Area Calculation
This solution explains how to find the area of a triangle when two sides and the angle between them are given. We are given a triangle ABC with specific measurements for two sides and the angle between them.
Given Information
- Angle A (\(\angle A\)): \(30^\circ\)
- Side AB: 7 cm
- Side AC: 12 cm
Area Formula for a Triangle
The area of a triangle can be calculated using the formula when two sides and the angle between them are known:
Area = \(\frac{1}{2} \times \text{side}_1 \times \text{side}_2 \times \sin(\text{included angle})\)
In this specific triangle ABC, the sides are AB and AC, and the angle between them is \(\angle A\). So the formula becomes:
Area of \(\triangle ABC = \frac{1}{2} \times AB \times AC \times \sin(\angle A)\)
Step-by-Step Calculation
Substitute the known values into the area formula:
Area = \(\frac{1}{2} \times AB \times AC \times \sin(\angle A)\)
Area = \(\frac{1}{2} \times 7 \, \text{cm} \times 12 \, \text{cm} \times \sin(30^\circ)\)
We know the value of \(\sin(30^\circ)\) is \(\frac{1}{2}\). Substitute this value into the equation:
Area = \(\frac{1}{2} \times 7 \times 12 \times \frac{1}{2}\)
Now, perform the multiplication:
Area = \(\frac{7 \times 12 \times 1}{2 \times 2}\)
Area = \(\frac{84}{4}\)
Area = \(21\)
The unit for the area is square centimeters (cm\(^2\)).
Area = \(21 \, \text{cm}^2\)
Final Result
The area of the triangle ABC is \(21 \, \text{cm}^2\).
Similar Questions
What is \(AB + BC\) equal to?
- In a right-angled triangle ABC, AB = 15 cm, BC = 20 cm and AC = 25 cm. Further, BP is the perpendicular on AC. What is the difference in the area of triangles PAB and PCB?
Let X, Y and Z be the midpoints of the sides BC, CA and AB of a triangle ABC respectively. Consider the following statements:
I. The quadrilateral AZXY is a parallelogram.
II. The area of the quadrilateral AZXY is half of the area of the triangle ABC.
Which of the statements given above is/are correct?- ABC is a triangle right angled at B. D is a point on AC such that BD is perpendicular to AC. If AB = \(p\) and BC = \(\sqrt{3}p\), then what is BD equal to?
- The sides of a triangle are 11 cm, 60 cm and 61 cm. What is the area of the triangle formed by joining the mid- points of the sides of the triangle?
Important Questions from Triangles
In a Δ ABC, if ∠A = 120° and AB = AC, then the values of ∠B and ∠C are respectively:
The product of the perimeter of a triangle, the radius of its in‐circle, and a number gives the area of the triangle. The number is
In ∆ABC, 2∠A = 3∠B = 6∠C. What is the value of the largest angle among these three angles?
The longest side in a right-angled triangle is called :
The sum of two sides of a triangle with the third side is
