Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized for You
we will pick new questions that match your level based on your Timer History
Track Your Progress
every week, we’ll send you an estimated GRE score based on your performance
Practice Pays
we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:
Your score will improve and your results will be more realistic
Is there something wrong with our timer?Let us know!
Attached pdf of this Article as SPOILER at the top! Happy learning!
Hi All,
I have recently uploaded a video on YouTube to discuss Similar Triangles in Detail:
Following is covered in the video
¤ Definition of Similar Triangles ¤ Properties of Similar Triangles ¤ Relationship of Perimeter of two Similar Triangles ¤ Relationship of Area of two Similar Triangles
Definition of Similar Triangles
Two triangles are similar if at least two of their corresponding angles are equal.
=> If two angles are equal then the third angle will also be equal (As sum of the angles is 180°) => If all three corresponding angles of two triangles are equal then they are similar triangles
Attachment:
Image-1.jpg [ 10.69 KiB | Viewed 1196 times ]
In above Figure △ ABC and △ DEF are similar because ∠A = ∠D, ∠B = ∠E and ∠C = ∠F
Properties of Similar Triangles
If two triangles are similar, then their corresponding sides will be in the same ratio.
Attachment:
Image-1.jpg [ 10.69 KiB | Viewed 1196 times ]
In above Figure △ ABC and △ DEF are similar => \(\frac{AB}{DE}\) = \(\frac{BC}{EF}\) = \(\frac{AC}{DF}\)
Relationship of Perimeter of two Similar Triangles
Ratio of Perimeter of two similar triangles is equal to the ratio of their sides.
Attachment:
Image-1.jpg [ 10.69 KiB | Viewed 1196 times ]
In above Figure △ ABC and △ DEF are similar => \(\frac{AB}{DE}\) = \(\frac{BC}{EF}\) = \(\frac{AC}{DF}\) = k (assume) => AB = k*DE => BC = k*EF => AC = k*DF
=> Perimeter of △ ABC / Perimeter of △ DEF = \(\frac{AB + BC + AC }{ DE + EF + DF}\) = \(\frac{k*DE + k*EF + k*DF }{ DE + EF + DF}\) = \(\frac{k * ( DE + EF + DF ) }{ DE + EF + DF}\) = k = \(\frac{AB}{DE}\) = \(\frac{BC}{EF}\) = \(\frac{AC}{DF}\)
Relationship of Area of two Similar Triangles
Ratio of Area of two similar triangles is equal to square of ratio of their sides.
Attachment:
Image-2.jpg [ 11.85 KiB | Viewed 1153 times ]
In above Figure △ ABC and △ DEF are similar and AG is perpendicular(⊥) to BC and DH ⊥ EF
If we consider △ AGB and △ DHE, then ∠B = ∠E, ∠G = ∠H = 90° => ∠GAB = ∠HDE => △ AGB and △ DHE => Their sides will be in the same ratio => \(\frac{AG}{DH}\) = \(\frac{GB}{HE}\) = \(\frac{AB}{DE}\) ...(1)
And we already know that △ ABC and △ DEF => \(\frac{AB}{DE}\) = \(\frac{BC}{EF}\) = \(\frac{AC}{DF}\) = k ...(2)
From (1) and (2) we get \(\frac{AG}{DH}\) = \(\frac{GB}{HE}\) = \(\frac{AB}{DE}\) = \(\frac{AB}{DE}\) = \(\frac{BC}{EF}\) = \(\frac{AC}{DF}\) = k
=> Area of △ ABC / Area of △ DEF = (\(\frac{1}{2}\) * BC * AG) / (\(\frac{1}{2}\) * EF * DH) = \(\frac{BC * AG }{ EF * DH}\) = \(\frac{BC}{EF}\) * \(\frac{AG}{DH}\) = k * k = \(k^2\)