What is the area of the triangle ABC with sides a = 10cm and c = 4cm angle B = 30°?
10 cm 2
Understanding the Problem: Finding the Area of Triangle ABC
The question asks for the area of a triangle ABC. We are given the lengths of two sides, a and c, and the measure of the angle B between them. This specific scenario calls for using a particular formula for the area of a triangle.
Formula for Triangle Area with Two Sides and Included Angle
When you know the lengths of two sides of a triangle and the measure of the angle between those two sides, you can calculate the area using the trigonometric formula:
\(\text{Area} = \frac{1}{2}ab \sin(C)\)
or similarly,
- \(\text{Area} = \frac{1}{2}bc \sin(A)\)
- \(\text{Area} = \frac{1}{2}ac \sin(B)\)
In this problem, we are given sides a and c, and the angle B, which is the angle included between sides a and c. So, we will use the formula:
\(\text{Area} = \frac{1}{2}ac \sin(B)\)
Applying the Area Formula to Triangle ABC
We are given the following values:
- Side a = 10 cm
- Side c = 4 cm
- Angle B = 30°
Now, we substitute these values into the formula:
\(\text{Area} = \frac{1}{2} \times 10 \text{ cm} \times 4 \text{ cm} \times \sin(30°)\)
Calculating the Sine Value
We need the value of \(\sin(30°)\). A common trigonometric value is \(\sin(30°) = \frac{1}{2} = 0.5\).
Completing the Area Calculation
Substitute the value of \(\sin(30°)\) into the equation:
\(\text{Area} = \frac{1}{2} \times 10 \times 4 \times 0.5\)
Perform the multiplication:
\(\text{Area} = \frac{1}{2} \times 40 \times 0.5\)
\(\text{Area} = 20 \times 0.5\)
\(\text{Area} = 10\)
The units for area are square centimeters (cm\({^2}\)).
So, the area of triangle ABC is 10 cm\({^2}\).
Comparing with the Options
Let's compare our calculated area with the given options:
- Option 1: 16 cm\({^2}\)
- Option 2: 12 cm\({^2}\)
- Option 3: 10 cm\({^2}\)
- Option 4: 8 cm\({^2}\)
Our calculated area of 10 cm\({^2}\) matches Option 3.
Revision Table: Triangle Area Formulas
| Method | Information Needed | Formula |
|---|---|---|
| Base and Height | Base (b) and corresponding height (h) | \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\) \(\text{Area} = \frac{1}{2}bh\) |
| Two Sides and Included Angle | Two sides (a, b) and the included angle (C) | \(\text{Area} = \frac{1}{2}ab \sin(C)\) (and variations for other sides/angles) |
| Three Sides (Heron's Formula) | Three sides (a, b, c) | First calculate the semi-perimeter, \(s = \frac{a+b+c}{2}\). Then, \(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\) |
Additional Information: Understanding Sine in Area Calculation
The trigonometric formula for the area of a triangle (\(\frac{1}{2}ab \sin(C)\)) is derived from the standard base-height formula. Consider a triangle with sides a, b, and angle C between them. If we take side 'a' as the base, the height 'h' relative to this base would be \(h = b \sin(C)\). Substituting this into the base-height formula (\(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\)), we get:
\(\text{Area} = \frac{1}{2} \times a \times (b \sin(C))\)
\(\text{Area} = \frac{1}{2}ab \sin(C)\)
This shows how the sine of the included angle relates the side lengths to the triangle's height and ultimately its area.
Similar Questions
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1. The points lie on a straight line.
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Important Questions from Area
For the equations \(ax + (a^2 + 1)y = 4\) and \(4x + ay = a^2\), which of the following statements is TRUE?
What is the area of the parallelogram?
Consider the following statements :
1. If in a triangle ABC, A = 2B and b = c, then it must be an obtuse-angled triangle.
2. There exists no triangle ABC with A = 40°, B = 65° and a/c = sin 40° cosec 15°Which of the above statements is/are correct?
Which one of the following is the area of a sector of a circle of radius 10 cm formed by an arc length of 15 cm?
Consider the following statements in respect of the points (p, p - 3), (q + 3, q) and (6, 3):
1. The points lie on a straight line.
2. The points always lie in the first quadrant only for any value of p and q.
Which of the above statements is/are correct?
