Math topic中文

Plane geometry proofs

use of:

properties of parallel lines cut by a transversal, perpendicular and angle bisectors, triangles, special quadrilaterals and circles
congruent and similar triangles
midpoint theorem
tangent-chord theorem (alternate segment theorem)

Practice questions

1The triangle ABC is such that its area is $\dfrac{16 + 7\sqrt{10}}{2}\ \text{cm}^2$ , the length of AB is $(3\sqrt{2} + \sqrt{5})\ \text{cm}$ …
2如果三角形的两条边分别为4和6，那么连结该三角形三边中点所得的周长可能是下列数据中的【】 A. 6 B. 8 C. 10 D. 12
3In $\Delta ABC$ , $AC = 12\text{ cm}$ , $\angle ABC$ is a right angle, $\angle ACB = \frac{\pi}{6}$ radians and $M$ is the…
4The diagram shows a quadrilateral $ABCD$ whose vertices lie on the circumference of a circle. The point $E$ lies on the…
5The equation of a circle is $x^2 + y^2 + 8x - 20y - 53 = 0$ . Find the radius of the circle and the coordinates of its ce…
6A circle with centre O passes through the points P(-1,7) and Q(0,8). State the relationship between the perpendicular bi…
7In the diagram below, TAX and TCY are tangents to the circle at A and C respectively. AC meets TD at E and D is on BC su…
8The diagram shows a circle passing through the vertices of a triangle $ABC$ . The tangent to the circle at $A$ meets $BC$ …
9A cone of height, $h$ and radius, $r$ fits exactly inside a sphere of radius $5\text{ cm}$ as shown below. $\left(\text{Volume of cone} = \frac{1}{3}\pi r^2 h\right)$ …
10The diagram shows two chords AB and CD intersect at X such that angle CXB = $90^\circ$ . Given that $AC = 5\ \text{cm}$ ,…
11The diagram shows two triangles DAB and DBC. It is given that BD = 5 m and BC = 2 m. Angle DAB = angle DBC = angle DEC =…
12The diagram below shows a circle with chords $PQ$ and $RS$ intersect at the point $X$ . $P, Q, R$ and $S$ are points on t…
13A(1, 7) and B(9, 7) lie on a circle $C_1$ with centre P. The line $4y + 3x = 31$ is a normal to the circle and passes th…
14In the diagram, $TQ$ and $PR$ are the diameters of the circle with centre $O$ . The tangent at $T$ meets $RP$ produced at…
15In the diagram, $PQS$ lies on a circle and $PCBO$ lies on another circle. PQ is a common chord of the two circles. BPS i…
16 $ABCD$ is a rectangle which fits inside a semicircle of radius 8 cm and centre $O$ . It is given that $AB = x$ cm and $BC = y$ …
17The diagram shows a point $A$ on the circle and $XAY$ is a tangent to the circle. Points $S$ , $B$ and $C$ lie on the cir…
18Solutions to this question by accurate drawing will not be accepted. In the trapezium $OABC$ , the point $A$ has coor…
19Solutions to this question by accurate drawing will not be accepted. The diagram, which is not drawn to scale, shows…
20The diagram shows a trapezium $ABCD$ . The point $X$ lies on line $AB$ such that $DX = 12\text{ cm}$ and $CX = 9\text{ cm}$ …
21The equation of a circle is $x^2 + y^2 + 2x - 8y - 8 = 0$ . Find the coordinates of the centre, $C$ and the radius of the…
22A cone of height 12 cm and base radius 5 cm is placed over a cylinder of radius $r$ cm and height $h$ cm. The cone is in…
23 $ABCD$ is a parallelogram. Point $A$ lies on the $y$ -axis and coordinates of $D$ is $(6, -1)$ . $AC$ is a horizontal line…
24The diagram shows a circle, centre $O$ , with diameter $AB$ . The points $D$ and $F$ lie on the circle. The point $E$ is s…
25[The volume of a cone of height $h$ and base radius $r$ is $\frac{1}{3}\pi r^2 h$ .] A hollow right circular cone of base…
26The diagram shows a prism in which the cross-section is a trapezium $ABCD$ . $AB = 4x\text{ cm}$ , $BC = DC = AD = 2x\text{ cm}$ …
27In the diagram below, $BD$ is a tangent to circle $ACB$ and $AB$ is a tangent to circle $BCDE$ . $ACFE$ is a straight lin…
28 $P$ , $Q$ , $R$ and $S$ are four villages located on an island, where $Q$ is due north of $R$ . $PQ = 924\text{ m}$ , $QR = 640\text{ m}$ …
29梯形的上底长为2cm，下底长为8cm，则梯形的中位线长为 cm．
30The diagram shows the front view of a rectangular block $ABCE$ , with dimensions 14 cm by 8 cm, placed on a ramp, $VE$ , t…
31A circle, with centre $C$ , has equation $x^2 + y^2 -10x -4y +25 = 0$ . Find the coordinates of $C$ and the radius of the…
32The diagram shows a pair of similar triangles $YUX$ and $VUW$ . Point $Y$ lies on the straight line $VU$ such that $VY = 12\text{ cm}$ …
33The figure shows three points $A$ , $C$ and $D$ whose coordinates are $(3, 5)$ , $(4, -2)$ and $(7, 2)$ respectively. Find…
34The diagram shows a quadrilateral $ABCD$ with vertices $A(-4, -3)$ , $B(6, 1)$ , $C(3, 5)$ and $D(-2, 3)$ . $M$ which lies…
35In the diagram, $AP$ is a tangent to the circle passing through $A$ , $B$ and $C$ . $AC$ is the diameter of the circle and…
36The figure shows a piece of cardboard in the shape of a trapezium in which $\angle BAD$ and $\angle ABC$ are right angle…
37The diagram shows a glass window $ABCDEF$ , consisting of a rectangle $ABEF$ of height $3x\text{ cm}$ and width $y\text{ cm}$ …
38Solutions to this question by accurate drawing will not be accepted. $ABCD$ is a trapezium where $AD$ is parallel to…
39In the diagram below, the line $OP$ makes an angle of $\alpha^\circ$ with the positive $x$ -axis such that $\tan \alpha = 0.2$ …
40Solutions to this question by accurate drawing will not be accepted. $OABC$ is a trapezium with right angle $OAB$ . T…
41The points $A(5, 7)$ and $B(-11, 15)$ lie on a circle $C$ . The centre of the circle $O$ lies on the line $y = -2x - 3$ .…
42In the diagram, A, B and C lie on a circle where O is the centre and AC is a diameter. The tangents to the circle at B a…
43The equation of a circle, $C_1$ is $x^2 + y^2 - 4x + 6y - 3 = 0$ . Find the coordinates of the centre and the radius of $C_1$ …
44等腰梯形的腰与中位线都是6厘米，则它的周长是厘米．
45The diagram shows two circles, $C_1$ and $C_2$ , intersecting at $A$ and at $B$ . $F$ is a point on $AB$ produced such tha…
46The diagram shows a play area for babies that is surrounded by partitions at PQ, QR and PS, where PQ = 18 m, PS = 40 m,…
47The diagram shows the side view of a $14\text{ cm}$ by $8\text{ cm}$ rectangular block $PQRS$ , placed on a ramp, $VS$ , t…
48In the diagram, $BC$ is a diameter of the circle. $ABC$ is a straight line and $AG$ is a tangent to the circle at point…
49The diagram shows a circle $ABCD$ and the tangent $ST$ to the circle at point $C$ . The point $B$ lies on $AT$ such that…
50The diagram below is not drawn to scale. In the diagram, D, E and F are points on the circle C1. Show that DE is the dia…
51In the diagram, $OC = 4\text{ m}$ , $OD = 7\text{ m}$ and angle $CBO = \text{angle } DOC = \text{angle } DAO = 90^\circ$ .…
52The equation of a circle with centre C is $x^{2}+y^{2}-8x-2y-152=0$ . Find the coordinates of C and the radius of the cir…
53The diagram shows two circles intersecting at $D$ and $F.$ ABC $is a tangent to the bigger circle at$ B $and$ BFH $is a tangent to the smaller circle at$ …
54A circle, $C_1$ , has the equation $x^2 + y^2 + 2x - 6y = 10$ . Find the coordinates of the centre and the exact radius of…
55Diagram 1 shows an equilateral triangle with sides of length 1 unit. An inverted white equilateral triangle with sides o…
56The diagram shows a trapezium $ABCD$ in which $AD$ is parallel to $BC$ and angle $ABC = 90^\circ$ . The coordinates of $A$ …
57In $\Delta ABC$ , $\angle ABC = 90^\circ$ , $AB = 12\text{ cm}$ and $AC = 13\text{ cm}$ . The rectangle $BPQR$ is such that…
58The diagram shows a lake (shaded) bordered by a straight level road AXB. Point C is on the surface of the lake and B is…
59The line $y = 10$ and $3y + 4x = 32$ are tangent to a circle $C$ at the points $(-2, 10)$ and $(2, 8)$ respectively. Sho…
60 $A$ , $B$ and $C$ are the angles of a triangle. Given that $\sin(A + B) = \frac{12}{13}$ . Find the exact value of $\tan C$ …