To find the area of the two triangles, let's discuss each one step-by-step.
### Triangle 1: Base and Height Given
Given:
- Base ([tex]\(b\)[/tex]) = 6 cm
- Height ([tex]\(h\)[/tex]) = 4 cm
Formula:
The area ([tex]\(A\)[/tex]) of a triangle when the base and height are given is:
[tex]\[A = \frac{1}{2} \times \text{base} \times \text{height}\][/tex]
Calculation:
Substitute the given values into the formula:
[tex]\[A = \frac{1}{2} \times 6 \times 4\][/tex]
[tex]\[A = \frac{1}{2} \times 24\][/tex]
[tex]\[A = 12\][/tex]
So, the area of the first triangle is:
[tex]\[12 \, \text{cm}^2\][/tex]
### Triangle 2: Sides Given
Given:
- Sides: [tex]\(a = 3 \, \text{cm}\)[/tex], [tex]\(b = 4 \, \text{cm}\)[/tex], [tex]\(c = 5 \, \text{cm}\)[/tex]
Formula:
To find the area of a triangle when the lengths of all three sides are given, we use Heron's formula, which is:
[tex]\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \][/tex]
where [tex]\(s\)[/tex] is the semi-perimeter of the triangle given by:
[tex]\[ s = \frac{a+b+c}{2} \][/tex]
Step-by-Step Calculation:
1. Calculate the semi-perimeter ([tex]\(s\)[/tex]):
[tex]\[ s = \frac{3 + 4 + 5}{2} \][/tex]
[tex]\[ s = \frac{12}{2} \][/tex]
[tex]\[ s = 6 \, \text{cm} \][/tex]
2. Substitute [tex]\(s\)[/tex], [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into Heron's formula:
[tex]\[ \text{Area} = \sqrt{6(6-3)(6-4)(6-5)} \][/tex]
[tex]\[ \text{Area} = \sqrt{6 \times 3 \times 2 \times 1} \][/tex]
[tex]\[ \text{Area} = \sqrt{36} \][/tex]
[tex]\[ \text{Area} = 6 \][/tex]
So, the area of the second triangle is:
[tex]\[6 \, \text{cm}^2\][/tex]
### Final Results
- Area of the first triangle: [tex]\(12 \, \text{cm}^2\)[/tex]
- Area of the second triangle: [tex]\(6 \, \text{cm}^2\)[/tex]
