3. Section 5.3
Exercise 135 Page 328
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Examining the diagram, we can identify two types of special triangles.
What do you know about the sides of these triangles?
Area≈ 21.86 units^2
Perimeter≈ 24.59units
We are going to find the area and perimeter of △ ABC.
To find the area of the triangle we need to know its base and height. Notice that the dashed segment is the triangle's height if AC is its base. This segment is made up of two different segments, which we will label x and y.
Examining the diagram, we see two right triangles: a 45-45-90 triangle and a 30-60-90 triangle. The former triangle is also an isosceles triangle, which means its legs are congruent.
In a 30-60-90 triangle the longer leg is always sqrt(3) times that of the shorter leg. With this information we know that y=4sqrt(3) and that the base of △ ABC is (4+4sqrt(3)). Now we have enough information to calculate the triangle's area.
To find the perimeter we need to know the length of the triangle's hypotenuse. In a 30-60-90 triangle the hypotenuse is twice the length of the shorter side. Because the shorter side is 4 units, the hypotenuse must be 8 units.
In a 45-45-90 triangle the hypotenuse is always sqrt(2) times the length of the legs. Therefore, this triangle has a hypotenuse of 4sqrt(2) units. We have all that we need to calculate the triangle's perimeter.
