Geometry Problems
With Solutions and Explanations for Grade 9
Detailed solutions and full explanations to Geometry Problems for grade 9 are presented.

Angles A and B are complementary and the measure of angle A is twice the measure of angle B. Find the measures of angles A and B,
Solution
Let A be the measure of angle A and B be the measure of angle B. Hence
A = 2B
Angles A and B are complementary; hence
A + B = 90°
But A = 2B; hence
2B + B = 90
3B = 90
B = 90 / 3 = 30°
A = 2B = 60°

ABCD is a parallelogram such that AB is parallel to DC and DA parallel to CB. The length of side AB is 20 cm. E is a point between A and B such that the length of AE is 3 cm. F is a point between points D and C. Find the length of DF such that the segment EF divide the parallelogram in two regions with equal areas.
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Solution
Let A1 be the area of the trapezoid AEFD. Hence
A1 = (1/2) h (AE + DF) = (1/2) h (3 + DF) , h is the height of the parallelogram.
Now let A2 be the area of the trapezoid EBCF. Hence
A2 = (1/2) h (EB + FC)
We also have
EB = 20  AE = 17 , FC = 20  DF
We now substitute EB and FC in A2 = (1/2) h (EB + FC)
A2 = (1/2) h (17 + 20  DF) = (1/2) h (37  DF)
For EF to divide the parallelogram into two regions of equal ares, we need to have area A1 and area A2 equal
(1/2) h (3 + DF) = (1/2) h (37  DF)
Multiply both sides by 2 and divide thm by h to simplify to
3 + DF = 37  DF
Solve for DF
2DF = 37  3
2DF = 34
DF = 17 cm

Find the measure of angle A in the figure below.
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Solution
A first interior angle of the triangle is supplementary to the angle whose measure is 129° and is equal to
180  129 = 51°
A second interior angle of the triangle is supplementary to the angle whose measure is 138° and is equal to
180  138 = 42°
The sum of all three angles of the triangle is equal to 180°. Hence
A + 51 + 42 = 180
A = 180  51  42 = 87°

ABC is a right triangle. AM is perpendicular to BC. The size of angle ABC is equal to 55 degrees. Find the size of angle MAC.
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Solution
The sum of all angles in triangle ABC is equal to 180°. Hence
angle ABC + angle ACM + 90° = 180°
Substitute angle ABC by 55 and solve for angle ACM
angle ACM = 180  90  55 = 35°
The sum of all angles in triangle AMC is equal to 180°. Hence
angle MAC + angle ACM + 90° = 180°
Substitute angle ACM by 35 and Solve for angle MAC
angle MAC = 180  90  angle ACM = 180  90  35 = 55°

Find the size of angle MBD in the figure below.
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Solution
The sum of all angles in triangle AMC is equal to 180°. Hence
56 + 78 + angle AMC = 180
angle AMC = 180  56  78 = 46°
Angles AMC and DMB are vertical angles and therefore equal in measures. Hence
angle DMB = 46°
The sum of angles of triangle DMB is equal to 180°. Hence
angle MBD + angle DMB + 62 = 180
Substitute angle DMB by 46 and solve for angle MBD.
angle MBD + 46 + 62 = 180
angle MBD = 180  46  62 = 72°

The size of angle AOB is equal to 132 degrees and the size of angle COD is equal to 141 degrees. Find the size of angle DOB.
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Solution
angle AOB = 132 and is also the sum of angles AOD and DOB. Hence
angle AOD + angle DOB = 132° (I)
angle COD = 141 and is also the sum of angles COB and BOD. Hence
angle COB + angle DOB = 141° (II)
We now add the left sides together and the right sides together to obtain a new equation.
angle AOD + angle DOB + angle COB + angle DOB = 132 + 141 (III)
Note that.
angle AOD + angle DOB + angle COB = 180°
Substitute angle AOD + angle DOB + angle COB in (III) by 180 and solve for angle DOB.
180 + angle DOB = 132 + 141
angle DOB = 273  180 = 93°

Find the size of angle x in the figure.
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Solution
The interior angle of the quadrilateral on the left that is supplementary to x is equal to
180  x
The interior angle of the quadrilateral on the left that is supplementary to the angle of measure 111° is equal to
180  111 = 69°
The sum of all interior angles of the quadrilateral is equal to 360°. Hence
41 + 94 + 180  x + 69 = 360
Solve for x
41 + 94 + 180  x + 69 = 360
384  x = 360
x = 384  360 = 24°

The rectangle below is made up of 12 congruent (same size) squares. Find the perimeter of the rectangle if the area of the rectangle is equal to 432 square cm.
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Solution
If the total area of the rectangle is 432 square cm, the area of one square is equal to
432 / 12 = 36 square cm
Let x be the side of one small square. Hence the area of one small circle equal to 36 gives
x^{2} = 36
Solve for x
x = 6 cm
The length L of the perimeter is equal to 4x and the width W is equal to 3x. Hence
L = 4 × 6 = 24 cm and W = 3 × 6 = 18 cm
The perimeter P of the rectangle is given by
P = 2 (L + W) = 2(24 + 18) = 84 cm

ABC is a right triangle with the size of angle ACB equal to 74 degrees. The lengths of the sides AM, MQ and QP are all equal. Find the measure of angle QPB.
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Solution
Angle CAB in the right triangle ACB is given by
90  74 = 16°
Sides AM and MQ in size and therefore triangle AMQ is isosceles and therefore
angle AQM = angle QAM = 16°
The sum of all interior angles in triangle AMQ is equal to 180°. Hence
16 + 16 + angle AMQ = 180
Solve for angle AMQ
angle AMQ = 180  32 = 148°
Angle QMP is supplementary to angle AMQ. Hence
angle QMP = 180  angle AMQ = 180  148 = 32°
Lengths of QM and QP are equal; hence triangle QMP is isosceles and therefore angle QPM is equal in size to angle QMP. Hence
angle QPM = 32°
Angle QPB is supplementary to angle QPM. Hence
angle QPM = 180  angle QPM = 180  32 = 148°

Find the area of the given shape.
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Solution
The area of the given shape may be found by subtracting the area of the right triangle (red) from the area of the large rectangle (see figure below).
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Sides of the right triangle (red) are given by
15  10 = 5 cm and 20  8 = 12 cm
Area of given shape = 20 × 15  (1/2) × 12 × 5 = 270 cm^{2}

Find the area of the shaded region.
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Solution
The area of the given shape may be found by subtracting the area of the rectangle at the top left from the area of the large rectangle.
Dimensions of the rectangle at top left
length = 30  8 = 22 cm , width = 15  4 = 11 cm
Area of given shape = 30 × 15  22 × 11 = 208 cm^{2}

The vertices of the inscribed (inside) square bisect the sides of the second (outside) square. Find the ratio of the area of the outside square to the area of the inscribed square.
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Solution
Let 2 x be the size of the side of the large square (see figure below).
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The area of the large square is
(2 x ) × ( 2x) = 4 x^{ 2}
The area of the inscribed square is
y × y = y ^{ 2}
Use of Pythagora's theorem gives
y ^{ 2} = x ^{ 2} + x ^{ 2} = 2 x^{2}
Ratio R of the area of the outside square to the area of the inside square is given by
R = 4 x^{ 2} / y ^{ 2} = 4 x^{ 2} / 2 x ^{ 2} = 4 / 2 = 2 / 1
Answers to the Above Questions
 measure of A = 60 degrees, measure of B = 30 degrees
 length of DF = 17 cm
 measure of A = 87 degrees
 size of angle MAC = 55 degrees
 size of angle MBD = 72 degrees
 size of angle DOB = 93 degrees
 size of angle x = 24 degrees
 perimeter of large rectangle = 84 cm
 measure of angle QPB = 148 degrees
 area of given shape = 270 square cm
 area of shaded region = 208 square cm
 ratio of area of outside square to area of inscribed square = 2:1

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