Question

In the given figure, ABC is a triangle (not on scale) in which AB =6cm,AC=8cm,BC=10cm and
AD is perpendicular to BC then AD = ?
(A) 23.04 cm
(B) 23.05 cm
(C) 23.06 cm
(D) 23.07 cm​

Answer 1

Answer:

The value of $AD$ is $4.8$ cm.

No option is correct.

Step-by-step explanation:

In Δ$ABC$, $AB$ = $6$ cm, $AC$ = 8 cm and $BC$ = 10 cm

Also, $AD$ is perpendicular to $BC$, i.e.,

Δ$ABD$ and Δ$ACD$ are right angled triangles.

Let $BD$ = $x$ cm. Then,

$CD$ = (10 - $x$) cm

In Δ$ABD$, by Pythagoras theorem,

⇒ $AB^2=AD^2+BD^2}$

⇒ $6^2=AD^2+x^2}$ (Given: AB = 6 cm)

⇒ $36=AD^2+x^2}$

⇒ $AD^2=36-x^2}$ . . . . . (1)

Now,

In Δ$ACD$, by Pythagoras theorem,

⇒ $AC^2=AD^2+CD^2}$

⇒ $8^2=AD^2+(10-x)^2}$ (Given: AC = 8 cm)

⇒ $64=AD^2+(100+x^2-20x)}$

⇒ $AD^2=64-100-x^2+20x}$

⇒ $AD^2=-36-x^2+20x$ . . . . . (2)

From (1) and (2), we get

$36-x^2 =-36-x^2+20x$

⇒ $36+36=20x$

⇒ $20x=72$

⇒ $x=72/20$

⇒ $x=3.6$ cm

This implies $BD=3.6$ cm and $CD=6.4$ cm.

Substitute the value $3.6$ for $x$ in the equation (1) as follows:

⇒ $AD^2=36-(3.6)^2}$

Simplify as follows:

⇒ $AD^2=36-12.96$

⇒ $AD^2=23.04$ cm

⇒ $AD=4.8$ cm.

Answer 2

Answer:

Length of AD = 4.8cm.

Step-by-step explanation:

Explanation:

Given in the question that ABC is a triangle.

AB = 6cm , AC = 8cm , BC = 10cm .

According to the question we need to find the value of AD

Let BD be x cm then DC = BC - BD = (10 -x)cm.

Step 1:

According to the question AD is perpendicular to BC.

Therefore, in right angle triangle ADB , ∠ADB = 90°

By Pythagoras theorem,

$AB^2 = BD ^2 + AD^2$

⇒$AD^2 = AB^2 - BD^2$

⇒$AD^2$ = $6^2 - x^2$ = 36 -$x^2$  ......(i)

Similarly, in right angle triangle ADC, ∠ADC = 90°

Again by Pythagoras theorem,

$AD^2 = AC^2 - DC^2$

⇒$AD^2 = 8^2 - (10-x)^2$

⇒$AD^2 = 64 - (100 +x^2 - 20x)$ .......(ii)

Step 2:

From (i) and (ii)

$36 - x^2$ = $64 - (100 +x^2 - 20x)$

⇒36$- x^2 = 64 - 100 - x^2 + 20x$

⇒36 = -36 + 20x

⇒20x = 36 + 36 = 72

⇒x = $\frac{72}{20}$ = 3.6cm

So, BD = 3.6cm

Step 3:

Now, put the value of x = 3.6cm in any one of the  equations.

Therefore,

$AD = \sqrt{36 - (3.6)^2}$

AD = $\sqrt{36 - (\frac{36}{10} )^2}$

⇒AD = $\sqrt{\frac{3600 - 1296}{100}}$ = $\sqrt{\frac{2304}{100} }$ = $\frac{48}{10}$ = 4.8cm

Final answer:

Hence, the value of AD is 4.8cm.

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