Question

OR
State Thales theorem and hence find the value of x in the given figure, in which DE||AB.​

Answer 1

Step-by-step explanation:

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Answer 2

Let's understand the BPT Concept :

Thales Theorem is also known as Basic Proportionality Theorem.

Let us now state the Thales Theorem which is as follows:

If a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio.

So,

According to the Thales Theorem,

Here, DE||AB and AD/DC = BE/EC (divided in same ratio)

We are asked to find the x.

Explanation :

According to the Thales Theorem,

$\Rightarrow\bf\dfrac{AD}{DC}=\dfrac{BE}{EC}$

where,

• AD = 3x + 19
• DC = x + 3
• BE = 3x + 4
• EC = x

Substituting the values,

$\\ \Rightarrow\sf\dfrac{3x+19}{x+3}=\dfrac{3x+4}{x}$

Cross Multiplying,

$\\ \Rightarrow\sf x(3x+19)=(3x+4)(x+3)$

Removing the brackets,

$\\ \Rightarrow\sf 3x^2+19x=3x^2+9x+4x+12$

Cancelling 3x² from both the sides,

$\\ \Rightarrow\sf \cancel{3x^2}\ +19x=\cancel{3x^2}\ +9x+4x+12$

$\\ \Rightarrow\sf19x=9x+4x+12$

$\\ \Rightarrow\sf19x=13x+12$

Transposing 13x to LHS,

$\\ \Rightarrow\sf19x-13x=12$

$\\ \Rightarrow\sf6x=12$

$\\ \Rightarrow\sf x=\dfrac{12}{6}$

$\\ \Rightarrow\sf x=\cancel{\dfrac{12}{6}}$

$\\ \therefore\boxed{\bf x=2.}$

The value of x is 2.

Explore More :

1) AAA similarity :

If two triangles are equiangular( all three angles are equal to each other), then they are similar.

Example :

In ΔABC and ΔDEF, ∠A = ∠D, ∠B = ∠E and ∠C= ∠F then ΔABC ~ ΔDEF by AAA criteria.

2) AA similarity :

If two angles of one triangle are respectively equal to tow angles of another triangle, then the two triangles are similar.

Example :

In ΔPQR and ΔDEF, ∠P = ∠D, ∠R = ∠F then ΔPQR ~ ΔDEF by AA criteria.

3) SSS similarity :

If the corresponding sides of two triangles are proportional, then the two triangles are similar.

Example :

In ΔXYZ and ΔLMN, XY = LM, YZ = MN and XZ = LN then ΔXYZ ~ ΔLMN by SSS criteria.

4) SAS similarity :

If in two triangles, one pair of corresponding sides are proportional and the included angles are equal then the two triangles are similar.

Example :

In ∆ABC & ∆PQR,∠A = ∠P, AB = PQ, AC = PR then ∆ABC ~ ∆PQR by SAS criteria.