Download Geometry Worksheet 1 – Angles and Circle Theorems

Geometry Worksheet 1 – Angles and Circle Theorems
All SMC, BMO and Mentoring problems are © UKMT (www.ukmt.org.uk)
1. [Source: SMC] ABCDEFGH is a regular octagon. P is the point inside the octagon such that
triangle ABP is equilateral. What is the size of angle APC?
A
90°
B
112.5° C
117.5°
D
120° E
135°
2. If two chords AB and CD intersect at a point X, prove that 𝐴𝑋 × 𝑋𝐵 = 𝐶𝑋 × 𝑋𝐷 (i.e. prove
the Intersecting Chord Theorem).
3. [Source: SMC] The size of each exterior angle of a regular polygon is one quarter of the size
of an interior angle. How many sides does the polygon have?
A
6
B
8
C
9
D
10
E
12
4. [Source: SMC] In the diagram AB, CB, and XY are tangents to the circle with centre O and
angle ABC = 48°. What is the size of angle XOY?
C
X
B
48°
O
Y
A
A
42°
B
69°
C
66°
D
48°
E
84°
5. [Source: SMC] In the figure shown, what is the sum of the interior angles at A, B, C, D, E?
A
90°
B
135° C
150°
D
180° E
more information required.
E
A
B
D
C
6. [Source: SMC] A point 𝑃 is chosen inside a square 𝐴𝐵𝐶𝐷. What is the probability that ∠𝐴𝑃𝐷
is acute? (Hint: your expression will involve 𝜋)
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7. The largest circle which it is possible to draw inside triangle 𝑃𝑄𝑅 touches the triangle at 𝑆, 𝑇
and 𝑈, as shown in the diagram. The size of ∠𝑆𝑇𝑈 = 55°. What is the size of ∠𝑃𝑄𝑅?
𝑄
𝑈
𝑆
𝑃
55°
𝑇
𝑅
8. [Source: SMC] In the diagram, 𝑂 is the centre of the circle, ∠𝐴𝑂𝐵 = 𝛼 and ∠𝐶𝑂𝐷 = 𝛽. What
is the size of ∠𝐴𝑋𝐵 in terms of 𝛼 and 𝛽?
A
D
1
𝛼
2
1
− 𝛽
2
180° − 𝛼 − 𝛽
B
E
1
1
90° − 𝛼 − 𝛽
2
2
More information needed.
C
𝛼−𝛽
9. [Source: UKMT Mentoring] If a 𝑛-sided polygon has exactly 3 obtuse angles (i.e. 90° < 𝜃 <
180°), then determine the possible values of 𝑛 (Hint: determine the possible range for the
sum of the interior angles, and use these inequalities to solve).
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Geometry Worksheet 1 - ANSWERS
1. ABCDEFGH is a regular octagon. P is the point inside the octagon such that triangle
ABP is equilateral. What is the size of angle APC?
C
P
B
A
B. ΔAPB is equilateral hence ABP = 60°
But BC = AB = PB so ΔPBC is isosceles with PBC  135  – 60  75 .
1
 BPC  (180 – 75) and APC  52.5  60 .
2
2. If two chords AB and CD intersect at a point X, prove that 𝑨𝑿 × 𝑿𝑩 = 𝑪𝑿 × 𝑿𝑫 (i.e. prove
the Intersecting Chord Theorem).
Clearly ∠𝐴𝑋𝐷 = ∠𝐶𝑋𝐵 (opposite angles are equal). Also, ∠𝐷𝐴𝐵 = ∠𝐵𝐶𝐷 (angles are in
𝐴𝑋
𝐶𝑋
same segment are equal). We thus have two similar triangles, and hence 𝑋𝐷 = 𝑋𝐵, leading to
the desired result.
3. The size of each exterior angle of a regular polygon is one quarter of the size of an interior
angle. How many sides does the polygon have?
D. Let the exterior angle be x°. Then x + 4x = 180 and therefore x = 36. As is the case in all
convex polygons, the sum of the exterior angles = 360° and therefore the number of sides =
360/36 = 10.
4. C. Let the points of contacts of the tangents be P Q and R as shown and let XOQ  x and
QOY  y . Then since OX is the axis of symmetry of the tangent kite OPXQ, it bisects
POQ so XOP  x .
P
C
X
48°
O
x
y
A
R
B
Q
Y
Similarly ROY  y . Thus, in the quadrilateral OPBR
we have
2x + 2y + 2 × 90 o + 48 o = 360 o
i.e. 2 (x + y) = 132 o
i.e. x + y = 66 o
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5.
A
E
X
e
y
a
x
Y
b
B
D
d
c
C
Let a, b, c, d, e, x and y represent the sizes in degrees of certain angles in the figure, as
shown and let the points of intersection of AD with EB and EC be X and Y respectively.
Angle EXY is an exterior angle of triangle XBD so x = b + d. Similarly, angle EYX is
an exterior angle of triangle YAC so y = a + c. In triangle EXY, e + x + y = 180, so
a + b + c + d + e = 180.
6. A point 𝑷 is chosen inside a square 𝑨𝑩𝑪𝑫. What is the probability that ∠𝑨𝑷𝑫 is acute?
Observe the below diagram:
A
D
B
C
If the chosen point was on the semi-circular arc, then ∠𝐴𝑃𝐷 = 90° (angle on circumference
from diameter of a circle is 90). We see can see that if the point is inside the semicircle, the
angle is obtuse, and acute outside it. If we let the radius of the semicircle be 1, then its area
is 𝜋/2 and the area of the square 4. The proportion of the square that is inside the
semicircle is therefore 𝜋/8, and thus the proportion outside it (where we’ll have an acute
𝜋
angle) 1 − 8 .
7. The largest circle which it is possible to draw inside triangle 𝑷𝑸𝑹 touches the triangle at
𝑺, 𝑻 and 𝑼, as shown in the diagram. The size of ∠𝑺𝑻𝑼 = 𝟓𝟓°. What is the size of ∠𝑷𝑸𝑹?
By the Alternate Segment Theorem, ∠𝑄𝑈𝑆 = 55°, as is ∠𝑄𝑆𝑈. Thus ∠𝑃𝑄𝑅 = 180° −
(2 × 55°) = 70°.
1
8. ∠𝐴𝐶𝐵 = 2 𝛼 (angle subtended by an arc at the centre of a circle is twice the angle
1
1
1
subtended at the circumference) and, similarly, ∠𝐶𝐴𝐷 = 2 𝛽. Therefore ∠𝐴𝑋𝐵 = 2 𝛼 − 2 𝛽
(the exterior angle of a triangle is equal to the sum of the two interior opposite angles).
9. [Source: UKMT Mentoring] If a 𝒏-sided polygon has exactly 3 obtuse angles (i.e. 𝟗𝟎° <
𝜽 < 𝟏𝟖𝟎°), then determine the possible values of 𝒏 (Hint: determine the possible range
for the sum of the interior angles, and use these inequalities to solve).
If 3 angles are obtuse, the sum of these, say 𝑂, has the range 270 < 𝑂 < 540. For the 𝑛 − 3
angles that are not obtuse (i.e. acute or right-angled), then the sum 𝐴 has the range: 0 <
𝐴 ≤ 90(𝑛 − 3). The total of the interior angles is 180(𝑛 − 3), so
270 < 180(𝑛 − 3) < 540 + 90(𝑛 − 3)
Solving 270 < 180(𝑛 − 3), we get 𝑛 > 3.5 and solving 180(𝑛 − 3) < 540 + 90(𝑛 − 3), we
get 𝑛 < 7. Thus 𝑛 = 4, 5 𝑜𝑟 6.
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