Math topic中文

Ratio theorem

Use of the ratio theorem in geometrical applications.

Practice questions

1A temporary isolation centre is built to manage the increasing number of COVID-19 cases. The roof takes the shape of a t…
2In the triangle shown below, one vertex is origin O, and the two other vertices are A and B where $\overrightarrow{OA}=\mathbf{a}$ …
3Referred to the origin $O$ , the points $A$ and $B$ are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$ …
4The equations of planes $p_1$ and $p_2$ are given by $p_1:\;\mathbf{r}\cdot \begin{pmatrix}\alpha\\7\\2\end{pmatrix} = \alpha + 20,\; \alpha \in \mathbb{R},$ …
5The position vectors of the points $A$ and $B$ relative to the origin $O$ are $\mathbf{a}$ and $\mathbf{b}$ respectively…
6Referred to the origin O, the position vectors of the points A and B are $i - j + 2k\quad\text{and}\quad 2i + 4j + k$ re…
7Referred to the origin $O$ , the points $A$ and $B$ are such that $\overrightarrow{OA} = a$ and $\overrightarrow{OB} = b$ …
8Three non-zero vectors $\mathbf{p}$ , $\mathbf{q}$ and $\mathbf{r}$ are such that $3\mathbf{p} \times \mathbf{q} = \mathbf{r} \times \mathbf{p}$ …
9Relative to the origin O, the position vectors of two points A and B are a and b respectively. The length of a is k unit…
10In the diagram, O is centre of the rectangular base ABCD of a right pyramid with vertex V. Perpendicular unit vectors i,…
11OABC is a rhombus. The points A and C are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OC}=\mathbf{c}$ …
12Relative to the origin $O$ , points $A$ , $B$ and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ and $\mathbf{c}$ re…
13Relative to the origin $O$ , the points $A$ , $B$ , and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ and $3\mathbf{a}+\mathbf{b}$ …
14With reference to the origin $O$ , the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectivel…
15The plane $p$ passes through the points $A$ , $B$ and $C$ with coordinates $(0, 1, 1)$ , $(2, -1, 4)$ and $(-2, -1, 0)$ re…
16With respect to the origin $O$ , the fixed points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respect…
17With reference to the origin $O$ , the position vectors of points $A$ , $B$ and $C$ are expressed as $\overrightarrow{OA}$ …
18A tetrahedron $OABC$ has four triangular faces with a vertex $O$ at the origin. The coordinates of $A$ , $B$ and $C$ are…
19Referred to the origin $O$ , the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively such…
20The diagram shows a pyramid DOABC. Taking unit vectors i, j, k as shown, the position vectors of A, B, C, D are given by…
21Relative to the origin $O$ , the position vectors of points $A$ and $B$ are $\mathbf{a}$ and $\mathbf{b}$ respectively, w…
22Relative to the origin $O$ , the points $A$ , $B$ , and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ and $3\mathbf{a}+\mathbf{b}$ …
23Referred to the origin O, let P, Q and R be distinct points with position vectors $\mathbf{p}$ , $\mathbf{q}$ and $\mathbf{r}$ …
24Two lines, $l_1$ and $l_2$ , have equations $\frac{x - 1}{2} = -z - 1, y = 2$ and $x + 3 = 4 - y = z + 2$ respectively. D…
25The plane $p_1$ has equation $x - y + z = 0$ and the plane $p_2$ has equation $x + y + z = 2$ . Find the acute angle betw…
26The above diagram shows a circle with radius $r$ units and centre $O$ . The points $A$ and $B$ on the circle are such tha…
27With reference to the origin $O$ , the points $A$ , $B$ and $C$ are such that $\overrightarrow{OA} = \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$ …
28The points $P$ and $Q$ have position vectors $3\mathbf{i}+4\mathbf{j}-\mathbf{k}$ and $5\mathbf{i}+7\mathbf{j}+6\mathbf{k}$ …
29A player is controlling an avatar in a virtual reality game where points $(x,y,z)$ are defined relative to a fixed point…
30Referred to the origin $O$ , the points $A$ and $B$ are such that $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ …
31Relative to the origin $O$ , the points $A$ and $C$ have position vectors $\mathbf{a}$ and $2\mathbf{b}$ such that $\mathbf{a} = 5p\mathbf{i} - 2p\mathbf{j} + 4p\mathbf{k} \quad \text{and} \quad \mathbf{b} = \mathbf{i} - 2\mathbf{j} + 2\mathbf{k},$ …
32Referred to the origin $O$ , the points $A$ , $B$ and $C$ have position vectors $3\mathbf{i} - \mathbf{j}$ , $5\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}$ …
33The line $l_1$ and the plane $\pi_1$ have equations $\mathbf{r} = \begin{pmatrix} 1 \\ -3 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ 2 \\ -2 \end{pmatrix}, \lambda \in \mathbb{R}$ …
34Referred to the origin $O$ , points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively such tha…
35With respect to the origin $O$ , the points $A,B,C,D$ and $E$ have coordinates $A(2,3,4),\; B(6,5,7),\; C(8,9,6),\; D(4,7,3)$ …
36Referred to the origin $O$ , the points $A$ and $B$ are such that $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ …
37The diagram above shows a cuboid $OABCDEFG$ with horizontal base $OABC$ where $OA = 6\text{ cm}$ and $AB = 4\text{ cm}$ .…
38Relative to the origin $O$ , the point $A$ has position vector $3\mathbf{i} + 7\mathbf{j} + \mathbf{k}$ . The plane $\pi$ …
39The diagram shows a unit circle with centre $O$ and diameter $AB$ . The point $C$ lies on the circumference of the circle…
40As shown in the above parallelogram $OACB$ , $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ . I…
41The points $P$ and $Q$ have position vectors $3\mathbf{i}+4\mathbf{j}-\mathbf{k}$ and $5\mathbf{i}+7\mathbf{j}+6\mathbf{k}$ …
42It is given that $\overrightarrow{OA} = 3\mathbf{i} + \mathbf{j} + 3\mathbf{k}$ and $\overrightarrow{OB} = 5\mathbf{i} - 4\mathbf{j} + 3\mathbf{k}$ …
43Relative to the origin $O$ , two points $A$ and $B$ have position vectors given by $\mathbf{a} = 4\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$ …
44The points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ with respect to origin $O$ , where $\mathbf{a}$ …
45Referred to the origin $O$ , the position vectors of points $A$ , $B$ and $C$ are $6\mathbf{i}$ , $4\mathbf{i}+2\mathbf{j}+4\mathbf{k}$ …
46Referred to the origin $O$ , the points $A$ and $B$ are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$ …
47Referred to the origin $O$ , the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, whe…
48With respect to an origin $O$ , the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively w…
49The curve $C$ has parametric equations $x = t^2 + 2, \quad y = t^3, \quad \text{where } t \in \mathbb{R}.$ Sketch $C$ .…
50Referred to the origin $O$ , the position vectors of three points $A$ , $B$ and $P$ are $\mathbf{a}$ , $\mathbf{b}$ and $\mathbf{a} + 5\mathbf{b}$ …