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?
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?- 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?
- 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?
- 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?
Important Questions from Triangles
Among the following options, which are NOT sides of a triangle?
In a Δ ABC, if ∠A = 120° and AB = AC, then the values of ∠B and ∠C are respectively:
In the equilateral Δ ABC, the base BC is trisected at D and E. The line through D, Parallel to AB, meets AC at F and the line through E parallel to AC meets AB at G. If EG and DF intersect at H, then what is the ratio of the sum of the area of parallelogram AGHF and the area of the Δ DHE to the area of the Δ ABC?
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
A man goes 24 m towards east and then 10 m towards north. How far is he away from his initial position?
