Solutions and detailed explanations to problems in sample 2.

What is the measure of angle x if L1 and L2 are parallel lines?
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Solution
Let y be the vertical angle to x. Since L1 and L2 are parallel, angle y and the angle with measure 130° are supplementary. Hence
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y + 130 = 180 or y = 180  130 = 50°
x and y are vertical angles and therefore have equal measures
x = 50°

Find x.
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Solution
The sum of all 3 angles of any triangle is equal to 180 degrees. Hence
(4x + 10) + (4x + 5) + (3x) = 180
Group like terms and solve for x
11x = 165 or x = 15

ABC is a triangle where AN is perpendicular to CB and BM perpendicular to AC. The length of BC is 10, that of AC is 12 and that of AN is 8. Find the length of BM.
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Solution
The area A of the given triangle may be calculated using the two altitutdes as follows
A = (1/2)(AN)(BC) or A = (1/2)(BM)(AC)
Hence
(1/2)(AN)(BC) = (1/2)(BM)(AC)
Multiply both sides by 2 and substitute known lengths
8 * 10 = MB * 12
Multiply both sides by 2 and substitute known lengths
BM = 80 / 12 = 6.7 (approximated to one decimal place)

The lengths of sides BA and BC of triangle ABC are equal. Find the measure of angle x.
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Solution
Since BA and BC have equal lengths, then the triangle is isosceles and the interior angles at A and C have equal measues which may be calculated as follows
A + C + 40 = 180 or 2A = 140 or A = 70°
The interior angle at A and angle x are supplementary. Hence
70 + x = 180 or x = 110°

Find the area of a right isosceles triangle with hypotenuse equal to 24.
Solution
The two legs of a right isosceles triangle have equal lengths; let x be one of these lengths. The area A of the triangle is given by
A = (1/2) x * x = (1/2)x^{2}
We now use Pythagora's theorem to find x as follows
x^{2} + x^{2} = 24^{2}
Simplify
2 x^{2} = 576
x^{2} = 288
We now calculate the area A as follows
A = (1/2)x^{2} = (1/2) 288 = 144

The total area of the shape below is equal to 240. Find the length of segment AC.
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Solution
Let us express the total area as the area of the upper and lower triangles with commom base AC
(1/2)(20)(AC) + (1/2)(28)(AC) = 240
Multiply all terms by 2, simplify and solve for AC
(20)(AC) + (28)(AC) = 480
48 AC = 480
AC = 480 / 48 = 10

The circle in the figure has all its vertices on the circle. The area of the square is 144. What is the area of the circle?
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Solution
Let x be the side of the square. The area is equal to x^{2}. Hence
x^{2} = 144 , solve for x, x = 12
The diameter d of the circle is equal to the length of the diagonal of the square. Using Pythagora's theorem, we obtain
x^{2} + x^{2} = d^{2}
Solve for d
144 + 144 = d^{2}
d = 12 √2
The radius r of the circle is equal to d/2
r = 12 √2 / 2 = 6 √2
The area A of the circle is given by
A = Pi r^{2} = 72 Pi

The measures, in degrees, of two consecutive interior angles M and N of a parallelogram are given by 2x + 10 and x + 20 respectively. Find the measures of the two angles.
Solution
Any two consecutive angle of a parallelogram are supplementary. Hence
(2x + 10) + (x + 20) = 180
Solve for x
3x + 30 = 180
x = 50
We now evaluate M and N as follows
M = 2x + 10 = 2(50) + 10 = 110°
N = x + 20 = 50 + 20 = 70°

The perimeter of a rectangle is 90 and its width is 10. What is its area?
Solution
Let L and W be the length and width of the rectangle. Using the formula of the perimeter, we write
2L + 2W = 90
W is given; hence
2L + 2(10) = 90
Solve for L
2L = 70 , L = 35
The area A of the rectangle is given by
A = L * W = 35 * 10 = 350

What is the ratio of the area of a square of side x to the area of a rectangle of width 2 x and length 3 x?
Solution
The area of a square of side x is x^{2} and the area of a rectangle of width 2x and length 3x is (2 x)(3 x) = 6 x^{2}. The ratio of the area of the square to the area of a rectangle is given by
x^{2} / (6 x^{2}) = 1 / 6
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