Question

1. m arc ce = 54 degree , m arc bd = 23 degree. find m of angle cae.
2. ab = 4.2, BC = 5.4, Ae = 12, find ad.
3. ab = 3.6, AC = 9 and ad = 5.4, find AE.

​

Answer 1

Given :

• m (arc CE) = 54°
• m ( arc BD) = 23°

To Find :

• Measure of $\angle$ CAE.

Solution :

$\angle$ CAE has its vertex outside the circle and intercepts arc CE and arc BD.

$\longrightarrow$ $\angle$ CAE = $\mathtt{\dfrac{1}{2}\:\times\:m\:(arc\:CE) -\:m(arc\:BD)}$

$\longrightarrow$ $\angle$ CAE = $\mathtt{\dfrac{1}{2}\:\times\:(54-23)}$

$\longrightarrow$ $\angle$ CAE = $\mathtt{\dfrac{1}{2}\:\times\:31}$

$\longrightarrow$ $\angle$ CAE = $\mathtt{15.5\:\degree}$

$\large{\boxed{\mathtt{\red{m\:\angle\:CAE\:=\:15.5\:\degree}}}}$

Solution 2 :

Given :

• AB = 4.2
• BC = 5.4
• AE = 12

To Find :

• Length of AD.

Solution :

AC = AB + BC ( A - B - C)

•°• AC = 4.2 + 5.4

•°• AC = 9.6

Since, chord CB and chord ED intersects at point A outside the circle.

•°• by theorem of external division of chords,

$\longrightarrow$ $\mathtt{AB\:\times\:AC\:=\:AD\:\times\:AE}$

$\longrightarrow$ $\mathtt{4.2\:\times\:9.6\:=\:AD\:\times\:12}$

$\longrightarrow$ $\mathtt{40.32\:=\:12AD}$

$\longrightarrow$ $\mathtt{AD\:=\:{\dfrac{40.32}{12}}}$

$\longrightarrow$ $\mathtt{AD\:=\:{\dfrac{4032}{1200}}}$

$\longrightarrow$ $\mathtt{AD\:={\dfrac{2016}{600}}}$

$\longrightarrow$ $\mathtt{AD\:=\:{\dfrac{672}{200}}}$

$\longrightarrow$ $\mathtt{AD\:=\:{\dfrac{336}{100}}}$

$\longrightarrow$ $\mathtt{AD\:=\:3.36}$

$\large{\boxed{\mathtt{\green{AD\:=\:3.36}}}}$

Solution 3 :

Given :

• AB = 3.6
• AC = 9
• AD = 5.4

To Find :

• Length of AE.

Solution :

Chord CB and chord ED intersects at point A outside the circle.

•°• by theorem of external division of chords,

$\longrightarrow$ $\mathtt{AB\:\times\:AC\:=\:AD\:\times\:AE}$

$\longrightarrow$ $\mathtt{3.6\:\times\:9\:=\:5.4\:\times\:AE}$

$\longrightarrow$ $\mathtt{\:32.4\:=\:5.4\:\times\:AE}$

$\longrightarrow$ $\mathtt{AE\:=\:{\dfrac{32.4}{5.4}}}$

$\longrightarrow$ $\mathtt{AE\:=\:{\dfrac{324}{54}}}$

$\longrightarrow$ $\mathtt{AE\:=\:6}$

$\large{\boxed{\mathtt{\blue{AE\:=\:6}}}}$