Question

In triangle ABC PQ parallel to BC and PQ:BC =1:3. Find the ratio of AP and PB

Answer 1

In triangle ABC if PQ//BC & PQ:BC = 1:3 then AP:PB = 1:2 .

Step-by-step explanation:

Given data:

In ∆ABC, PQ//BC and PQ:BC = 1:3

To find:

AP:PB

Solution:

In ∆APQ and ∆ABC, we have

∠A = ∠A ….. [common angles to both the triangles]

∠APQ = ∠ABC ……. [corresponding angles since PQ is given parallel to BC]

∴ By AA similarity, ∆APQ ~ ∆ABC

Since we know that the corresponding sides of two similar triangles are proportional to each other, so, we have

$\frac{AP}{AB}$ = $\frac{PQ}{BC}$

⇒ $\frac{AP}{AP+PB}$  = 1/3 …… [$\frac{PQ}{BC}$ = 1:3 given and AB = AP+PB]

⇒ 3AP = AP + PB

⇒ 3AP – AP = PB  

⇒ 2AP = PB

⇒ $\frac{AP}{PB}$ = ½

⇒ AP : PB = 1 : 2

Thus, the ratio of AP and PB is 1 : 2.

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Learn More:

If areas of 2 similar triangles are equal prove that they are congruent

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Proof of the theorem: ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

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Answer 2

Given that,

In triangle ABC, PQ parallel to BC.

The ratio of PQ : BC = 1:3

We know that,

The corresponding sides of two similar triangles are proportional to each other.

We need to calculate the ratio of AP and PB

Using diagram

In triangle APQ and triangle ABC,

Common angles to both the triangles

$\angle A=\angle A$

Corresponding angles

$\angle{APQ}=\angle{ABC}$

So, by AA similarity,

$\triangle APQ\sim\triangle ABC$

So, $\dfrac{AP}{AB}=\dfrac{PQ}{BC}$

Put the value into the formula

$\dfrac{AP}{AP+PB}=\dfrac{1}{3}$

$3AP=AP+PB$

$3AP-AP=PB$

$2AP=PB$

$\dfrac{AP}{PB}=\dfrac{1}{2}$

Hence, The ratio of AP and PB is 1:2