Question

What is the value of x in the given figure ? Answer with solution.​

Answer 1

ANSWER:

x = 8 cm

STEP-BY-STEP EXPLANATION:

Given that AD is perpendicular on BC of ∆ABC and ∠BAC = 90°, BC = 16 cm, DC = 4cm and AC = x.

We need to find out the value of x.

In right angled ∆ADC,

By Pythagoras theorem

H² = P² + B²

(Where H is hypotenuse having value x, P is perpendicular and B is base having value 4 cm.)

Let's say that value of perpendicular is y.

Substitute the values,

⇒ x² = y² + (4)²

⇒ x² = y² + 16 --------(eq 1)

In ∆ADB and ∆ADC

∠BAD = ∠DAC = 90°

∠ADB = ∠ADC = 90° (As AD is perpendicular to BC)

∆ADB ~ ∆ADC

Now,

⇒ DC/AC = AC/BC

Substitute the values,

⇒ 4/x = x/16

⇒ x² = 16(4)

⇒ x² = 64

⇒ x = √64

⇒ x = 8

Therefore, the value of x is 8 cm.

Answer 2

$\large\underline{\sf{Solution-}}$

Given that,

Triangle ABC is right angled triangle right-angled at A.

AD is drawn perpendicular to BC intersecting BC at D.

DC = 4 cm

BC = 16 cm

AC = x cm

Now,

$\rm \: In \: \triangle\:BAC \: and \: \triangle\:ADC$

$\rm \: \angle\:BAC \: = \: \angle\:ADC \: \: \: \{each \: 90 \degree \}$

$\rm \: \angle\:ACB \: = \: \angle\:DCA \: \: \: \{common \}$

$\rm\implies \:\triangle\:BAC \: \sim \: \triangle\:ADC \: \: \: \{AA \: similarity \}$

$\rm\implies \:\dfrac{BC}{AC} = \dfrac{AC}{DC}$

$\rm \: \dfrac{16}{x} = \dfrac{x}{4}$

$\rm \: {x}^{2} = 64$

$\rm\implies \:x = 8 \: cm \: \: \: \{ \: as \: x \: \cancel{ < } \: 0 \}$

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SHORT CUT TRICK

If a triangle ABC is right angled triangle right-angled at A and AD is drawn perpendicular to BC intersecting BC at D, then the following holds

$\rm \: {AC}^{2} = DC \times BC$

$\rm \: {BA}^{2} = BD \times BC$

$\rm \: {AD}^{2} = BD \times DC$

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ADDITIONAL INFORMATION

1. Pythagoras Theorem :-

This theorem states that : In a right-angled triangle, the square of the longest side is equal to sum of the squares of remaining sides.

2. Converse of Pythagoras Theorem :-

This theorem states that : If the square of the longest side is equal to sum of the squares of remaining two sides, angle opposite to longest side is right angle.

3. Area Ratio Theorem :-

This theorem states that :- The ratio of the area of two similar triangles is equal to the ratio of the squares of corresponding sides.

4. Basic Proportionality Theorem :-

This theorem states that : If a line is drawn parallel to one side of a triangle, intersects the other two lines in distinct points, then the other two sides are divided in the same ratio.