Example Geometry Problem #3

**Example Geometry Problem #3**

The problem states that the triangle depicted is an isosceles triangle. Remember, there are two qualities that define an isosceles triangle:

1) two identical sides

2) two identical angles

If you have one, you will automatically have the other. The other important fact about isosceles triangles that you need to remember for this question is that the sides that are identical are the two sides that are opposite the two angles that are the same (and vice-versa).

OK, so how does this information help us solve the problem? Well, we know that the two bottom angles are equal, because they are both labeled y. This means that the two sides of the triangle (not the bottom side) are equal because they are opposite the two identical angles. Since they’re equal, we can say

2x – 2 = 3x – 8 [subtract 2x from both sides]

-2 = x – 8 [add 8 to both sides]

6 = x

Now we know that x is 6.

The question is asking about the area of the triangle, which is given by the equation

A_{triangle} = ½bh

where b is the length of the base, and h is the height. From the diagram, we can see that the height is simply x, which we know to be 6. The base is defined to be

3x – 2

so we simply substitute 6 for x, and evaluate.

3x – 2 = 3(6) – 2

3x – 2 = 16

So the base of the triangle is 16, and the height is 6; thus, the area is

A_{triangle} = ½(16)(6) = 48

The answer is (C).

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Example Geometry Problem #4

**Example Geometry Problem #4**

The area of a circle is given by the equation

A_{circle} = πr^{2}

So the area of a semicircle is half of this

A_{semicircle} = ½πr^{2}

The areas of the three semicircles are given in the problem:

A_{1} = 8π = ½πr_{1}^{2}

A_{2} = 25π/2 = ½πr_{2}^{2}

A_{3} = 18π = ½πr_{3}^{2}

So if we solve for the radius in each of these cases, we get

8π = ½πr_{1}^{2}

25π/2 = ½πr_{2}^{2}

18π = ½πr_{3}^{2}

[multiply all three equations by 2 on both sides]

16π = πr_{1}^{2}

25π = πr_{2}^{2}

36π = πr_{3}^{2}

[divide all three equations by π on both sides]

16 = r_{1}^{2}

25 = r_{2}^{2}

36 = r_{3}^{2}

[take the square root of both sides on all three equations]

4 = r_{1}

5 = r_{2}

6 = r_{3}

Now remember, these are the radii, not the diameters! In order to find the perimeter of the triangle, which is what the question is asking for, we need to multiply each of these by 2 to get the diameters of the semicircles (which are also the sides of the triangle). Thus, the perimeter is

P_{triangle} = 2(4) + 2(5) + 2(6) = 30

The answer is (D).

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