Find diameter of circle using tangent properties by various method

Question Sample Titled 'Find diameter of circle using tangent properties by various method'

In the figure, ${A}{B}$ and ${A}{C}$ are the tangents to the circle at ${B}$ and ${C}$ respectively. ${B}{D}$ is a diameter of the circle. ${A}{C}$ produced and ${B}{D}$ produced meet at ${E}$ . If ${A}{B}={12}$ $\text{cm}$ and ${A}{E}={20}$ $\text{cm}$ , then ${B}{D}=$

${A}$${B}$${C}$${D}$${E}$
A
${12}$ $\text{cm}$ .
B
${6}$ $\text{cm}$ .
C
${11}$ $\text{cm}$ .
D
${9}$ $\text{cm}$ .

 Method ${1}$ : Using equal areas of triangles Let ${O}$ and ${r}$ $\text{cm}$ be the centre and radius respectively. Join ${O}{C}$ and ${O}{A}$ . ${O}{C}\bot{A}{E}$ and ${E}{B}\bot{A}{B}$ tangent ⊥ radius ${A}{C}$ $={A}{B}={12}$ $\text{cm}$ tangent properties ${E}{B}$ $=\sqrt{{{20}^{{2}}-{12}^{{2}}}}={16}$ $\text{cm}$ Pyth. theorem Area of $\triangle{A}{E}{B}$ $={(}$area of $\triangle{O}{B}{A}{)}+{(}$area of $\triangle{O}{A}{E}{)}$ $\dfrac{{1}}{{2}}{\left({12}\right)}{\left({16}\right)}$ $=\dfrac{{1}}{{2}}{\left({12}\right)}{\left({r}\right)}+\dfrac{{1}}{{2}}{\left({20}\right)}{\left({r}\right)}$ ${r}$ $={6}$ ∴  ${B}{D}={12}$ $\text{cm}$

${A}$${B}$${D}$${E}$${C}$${12}$ cm${20}$ cm${16}$ cm${r}$${r}$${O}$Method ${1}$

 Method ${2}$ : Considering smaller right-angled $\triangle{O}{C}{E}$ Let ${O}$ and ${r}$ $\text{cm}$ be the centre and radius respectively. Join ${O}{C}$ and ${O}{A}$ . ${O}{C}\bot{A}{E}$ and ${E}{B}\bot{A}{B}$ tangent ⊥ radius ${A}{C}$ $={A}{B}={12}$ $\text{cm}$ tangent properties ${E}{B}$ $=\sqrt{{{20}^{{2}}-{12}^{{2}}}}={16}$ $\text{cm}$ Pyth. theorem ${E}{C}$ $={20}-{12}={8}$ $\text{cm}$ ${E}{O}$ $={\left({16}-{r}\right)}$ $\text{cm}$ Consider $\triangle{O}{C}{E}$ , ${8}^{{2}}+{r}^{{2}}$ $={\left({16}-{r}\right)}^{{2}}$ Pyth. theorem ${\left({16}-{r}\right)}^{{2}}-{r}^{{2}}$ $={64}$ ${\left({16}-{r}+{r}\right)}{\left({16}-{r}-{r}\right)}$ $={64}$ ${16}{\left({16}-{2}{r}\right)}$ $={64}$ ${r}$ $={6}$ ∴  ${B}{D}={12}$ $\text{cm}$

${A}$${B}$${D}$${E}$${C}$${12}$ cm${20}$ cm${12}$ cm${8}$ cm${16}$ cm${r}$${r}$${O}$Method ${2}$

專業備試計劃

Level 4+ 保證及 5** 獎賞

ePractice 會以電郵、Whatsapp 及電話提醒練習

ePractice 會定期提供溫習建議

Level 5** 獎勵：會員如在 DSE 取得數學 Level 5** ，將獲贈一套飛往英國、美國或者加拿大的來回機票，唯會員須在最少 180 日內每天在平台上答對 3 題 MCQ。

Level 4 以下賠償：會員如在 DSE 未能達到數學 Level 4 ，我們將會全額退回所有會費，唯會員須在最少 180 日內每天在平台上答對 3 題 MCQ。

FAQ

ePractice 是甚麼？

ePractice 是一個專為中四至中六而設的網站應用程式，旨為協助學生高效地預備 DSE 數學（必修部分）考試。由於 ePractice 是網站應用程式，因此無論使用任何裝置、平台，都可以在瀏覽器開啟使用。更多詳情請到簡介頁面。

ePractice 可以取代傳統補習嗎？

1. 會員服務期少於兩個月；或
2. 交易額少於 HK\$100。

Initiating...

HKDSE 數學試題練習平台