Answer:
(a.) Area = 6.00 square units
(b.) Area = 15.59 square units
Explanation:
(a.) The regular formula for the area of a triangle is A = 1/2bh, where
- A is the area in square units,
- b is the base,
- and h is the height.
In this triangle, the height is 3 units and the base is 4 units. Thus, we plug in 3 for h and 4 for b in the formula to find A, the area in square units:
A = 1/2(4)(3)
A = 2 * 3
A = 6
Thus, the area of the triangle is 6.00 square units.
(b.)
- Before we can find the area of the triangle, we'll need to find the height.
- The altitude (a line extending from the vertice to the base of the triangle) is the height)
- Because this is an equilateral triangle with three congruent sides, the altitude splits the base into two congruent parts, whose lengths are 3 units since 3 + 3 = 6.
- The altitude is perpendicular to the base and creates a right triangle embedded in the larger triangle.
- Thus, we have a right triangle, where the altitude/height is one side, the 3-unit side is another side, and the 6-unit side is the hypotenuse.
- Now we can find the height of this embedded triangle using the Pythagorean theorem, which is
a^2 + b^2 = c^2, where
- a and b are the shorter sides called legs,
- and c is the longest side called the hypotenuse.
Thus, we can plug in 3 for a and 6 for c, allowing us to solve for b, the height of the entire triangle:
3^2 + b^2 = 6^2
9 + b^2 = 36
b^2 = 27
√(b^2) = √(27)
b = √(9 * 3)
b = 3√3 (leaving the answer in the simplest radical form will allow us to get a more exact answer when finding the area of the triangle.
Thus, the height of the triangle is 3√3 units.
Area of the triangle in (b.):
Now, we can plug in 6 for b and 3√3 for h in the triangle area formula to find A, the area of the triangle in square units:
A = 1/2(6)(3√3)
A = 3(3√3)
A = 9√3
A= 15.58845727
A = 15.59
Thus, the area of the triangle is about 15.59 square units.
I attached a picture that shows how the triangle in (b.) can be divided into two triangles.