Question

✯ Moderaters
✯ Brainly stars
✯ Other best users

Q. Fɪɴᴅ ᴛʜᴇ ʟᴇɴɢᴛʜ ᴏғ sɪᴅᴇ AB ɪɴ ᴛʜᴇ ғɪɢᴜʀᴇ . Rᴏᴜɴᴅ ʏᴏᴜʀ ᴀɴsᴡᴇʀ ᴛᴏ 3 sɪɢɴɪғɪᴄᴀɴᴛ ᴅɪɢɪᴛs.

$\leadsto\sf \pink { Don't \: spam \: please} \\ \leadsto\sf \pink{thanks} \: \red {♡} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$
​

Answer 1

Answer:

7.17

Step-by-step explanation:

Given :

• ∠B = 58°
• ∠BCA = 32°
• ∠D = 70°
• AD = 10
• DC = 10

To find :

Length of AB to 3 significant figures

Solution :

In ΔABC, by observation, we see that,

∠B + ∠C = 90°

⇒ 58° + 32° = 90°

⇒ 90° = 90°

So, ΔABC is a right angle triangle

Now, we know that,

$\boxed{\mathrm{tan\theta = \dfrac{opposite}{adjacent}}}$

Now, for 32°,

$\implies \mathrm{tan 32^{\circ} = \dfrac{AB}{AC}}$

$\implies \mathrm{AB = ACtan32^{\circ}}$ ___ [1]

Now, in ΔADC, 2 sides are of equal length. So, ΔADC is a isosceles triangle.

Let us draw a perpendicular from D to line AC and intersection point be named as M. [Image in attachement]

Now, as it is isosceles triangle, from it's properties, it bisects the angle 70° to 35° and 35°and also bisects AC to AM and MC of equal length.

We know that,

$\boxed{\mathrm{sin\theta = \dfrac{opposite}{hypotenuse}}}$

Now, in ΔADM,

for 35°, AD is hypotenuse, AM is opposite side

So,

$\implies \mathrm{sin35^{\circ} = \dfrac{AM}{AD}}$

$\implies \mathrm{AM = ADsin35^{\circ}}$

As AM is half of AC,

$\implies \mathrm{\dfrac{1}{2}AC = ADsin35^{\circ}}$

$\implies \mathrm{AC = 2(10)sin35^{\circ}}$

So, substituting the value of AC in [1], we get,

$\implies \mathrm{AB = (20sin35^{\circ})(tan32^{\circ})}$

We know that,

$\boxed{\begin{array}{cc}\mathrm{sin35^{\circ} = 0.57357}\\\mathrm{tan32^{\circ} = 0.62487}\end{array}}$

Substituting these values, we get,

$\implies \mathrm{AB = (20)(0.57357)(0.62487)}$

$\implies \mathrm{AB = 7.16812}$

We need to round up till 3 significant digits, so, we must have 1 digit before decimal point and 2 digits after decimal.

As, there is 8 which is >5 after 6, we need to round up 6 to 7.

Thus,

$\implies \mathrm{AB = 7.17}$

$\therefore \underline{\boxed{\Large{\textbf{\green{AB = 7.17}}}}}$

ᴛʜᴇ ʟᴇɴɢᴛʜ ᴏғ sɪᴅᴇ AB ɪɴ ᴛʜᴇ ғɪɢᴜʀᴇ Rᴏᴜɴᴅᴇᴅ ᴛᴏ 3 sɪɢɴɪғɪᴄᴀɴᴛ ᴅɪɢɪᴛs ɪs 7.17

Hope it helps!!

Answer 2

$\sf\huge\bold{Answer:}$

Given :

• ∠ABC=58°
• ∠ACB=32°
• ∠ADC=70°
• AD=DC=10 units

To find :

Length of side AB to 3 significant digits.

Solution :

Sum of all angles of triangle is 180°

In ∆ABC

∠ABC+∠ACB+∠BAC=180°

58°+32°+∠BAC=180°

90°+∠BAC=180°

∠BAC=180°-90°

∠BAC=90°

Hence, ∆ABC is right angled at A.

We know,

$\tt\red{ \tan( \theta) = \frac{opposite \: side \: of \: angle}{adjacent \: side \: of \: angle}}$

To find AB

$\tt \tan(32 \degree) = \frac{AB}{AC}$

$\tt\implies \: AB = \tan(32 \degree) AC...(i)$

In ∆ADC

AD=DC=10units

Since, two sides of a triangle are equal, it forms an isosceles triangle.

Construction : Draw a perpendicular DM to the line AC.

• AC is bisected by DM to two equal parts AM and MC → AM=MC=½AC

• ∠ADC=70° is bisected by DM to two equal angles of 35°

[properties of isosceles triangle]

Also,

$\tt\red{ \sin( \theta) = \frac{opposite \: side \: of \: angle}{hypotenuse \: of \: triangle} }$

To find AM

$\tt \: \sin(35 \degree) = \frac{AM}{AD}$

$\tt \: \implies \: AM = \sin(35 \degree) AD...(ii)$

As discussed above

AM=½AC

$\tt\implies \: \frac{1}{2} AC = \sin(35 \degree) AD$

$\tt\implies \: AC = 2 \times \sin(35 \degree) AD$

$\tt\implies \: AC = 2 \times \sin(35 \degree) \times 10$

$\tt\implies \: AC = 20 \sin(35 \degree)$

Now, substituting AC in (i)

$\tt\implies \: AB =( 20 \sin35 \degree) ( \tan32 \degree)$

Now,

$\tt\purple{ \sin(35 \degree) = 0.57357 }\: \\ \tt\purple{ \tan(32 \degree) = 0.62487}$

substituting these values, we get

$\tt\implies \: AB =( 20 ) \pink {(0.57357) (0.62487 )}$

$\tt\implies \: AB =( 20 ) {(0.57357) (0.62487 )}$

$\tt\implies \: AB =7.16812$

Rounding off to 3 significant digits

$\tt\implies \: AB =7.17$

• As we know 8>5 , hence 1 will add up to the last number.

$\huge \mathrm\red{ AB = 7.17}$