Geometry (Lines, Angles, Triangles, Polygons) Questions and Answers

Answer: B

The values are x = 30° and y = 40°.

RP passes through the centre, so it is a diameter, and an angle in a semicircle is a right angle. So angle RSP = 90° and angle RQP = 90°.

\[ \begin{aligned} \text{In } \triangle RSP: y &= 180^\circ - 90^\circ - 50^\circ = 40^\circ \\ \text{In } \triangle RQP: x &= 180^\circ - 90^\circ - 60^\circ = 30^\circ \end{aligned} \]

The first option gives the same two numbers in the wrong order (x = 40°, y = 30°), the commonest mistake here. 50° and 40° reuse the given angle instead of working out x.

Answer: D

A parallelogram with perpendicular diagonals is a rhombus.

The diagonals of a parallelogram bisect each other. If they are also perpendicular, each diagonal is the perpendicular bisector of the other, which makes all four sides equal: the parallelogram is a rhombus.

A square also has perpendicular diagonals, but it is a special rhombus with right angles, which the condition does not require; the general answer is rhombus. A rectangle's diagonals are equal but not perpendicular unless it is a square, and a trapezium is not a parallelogram.

Answer: C

Angle B is 40°.

Since AB = AC, the triangle is isosceles and the base angles B and C are equal. The angles of a triangle sum to 180°:

\[ \begin{aligned} \angle B + \angle C &= 180^{\circ} - 100^{\circ} = 80^{\circ} \\ \angle B &= \frac{80^{\circ}}{2} = 40^{\circ} \end{aligned} \]

50° comes from halving 100° instead of the remaining 80°, and 60° and 45° would not give a total of 180° with equal base angles.

Answer: A

The hypotenuse is 4 cm.

A right-angled isosceles triangle with equal sides \(a\) has hypotenuse \(a\sqrt{2}\), so

\[ \begin{aligned} 2a + a\sqrt{2} &= 4 + 4\sqrt{2} \\ a\sqrt{2}(\sqrt{2} + 1) &= 4(1 + \sqrt{2}) \\ a\sqrt{2} &= 4 \end{aligned} \]

Since \(a\sqrt{2}\) is the hypotenuse, it is 4 cm, and each leg is \(2\sqrt{2}\) cm. Check: \(2\sqrt{2} + 2\sqrt{2} + 4 = 4\sqrt{2} + 4\).

\(2\sqrt{2}\) cm is the length of each equal side, not the hypotenuse. 5 cm and 6 cm do not satisfy the perimeter.

Answer: A

The street is about 15.2 m wide.

The ladder's foot stays at one point, so the street's width is the sum of the two horizontal distances from the foot to each wall:

\[ \begin{aligned} \text{side 1} &= \sqrt{13^2 - 8^2} \\ &= \sqrt{105} \approx 10.25 \\ \text{side 2} &= \sqrt{13^2 - 12^2} = 5 \\ \text{width} &\approx 15.25 \text{ m} \end{aligned} \]

The paper's 15·2 m is this value cut to one decimal. 14 m and 14·6 m are both shorter than 10.25 + 5, and 12 m is the height of the second window, not a width.

Answer: B

The distance is 13 cm.

A tangent is perpendicular to the radius at the point of contact, so the radius, the tangent and the line from P to the centre form a right-angled triangle. The radius is half the diameter, 5 cm:

\[ OP = \sqrt{5^{2} + 12^{2}} = \sqrt{169} = 13\ \text{cm} \]

Using the diameter, 10 cm, as a side instead of the radius gives about 15.6 cm, near the 15 cm option; 12 cm is just the tangent length, and 10 cm the diameter.

Answer: D

Let the smaller side be b and the diagonal d = 3b. The other side l satisfies \[l^{2}=d^{2}-b^{2}=9b^{2}-b^{2}=8b^{2},\] so \(l=2\sqrt{2}\,b\).

The ratio of length to breadth is \(2\sqrt{2}:1\), option D. The ratio \(3:1\) is diagonal to side, \(\sqrt{3}:1\) and \(\sqrt{2}:1\) do not satisfy the Pythagoras relation here.

Check with numbers: if the smaller side is 1, the diagonal is 3 and the larger side is the square root of 8, which is 2 times the square root of 2, so the ratio is 2.828 to 1.

Answer: D

Each interior angle of a regular octagon is 135°.

The sum of the interior angles of an \(n\)-sided polygon is \((n - 2) \times 180^\circ\). For an octagon, \(n = 8\).

\[ \begin{aligned} \text{sum} &= 6 \times 180^\circ = 1080^\circ \\ \text{each} &= 1080^\circ \div 8 = 135^\circ \end{aligned} \]

Quicker: each exterior angle is \(360^\circ \div 8 = 45^\circ\), and \(180^\circ - 45^\circ = 135^\circ\). 108° is the angle of a regular pentagon, 120° of a hexagon and 150° of a regular 12-sided polygon.