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Math: Triangles 21 Oct 2019, 04:34
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Bunuel





• \(m=\sqrt{\frac{2b^2+2c^2-a^2}{4}}\), where \(a\), \(b\) and \(c\) are the sides of the triangle and \(a\) is the side of the triangle whose midpoint is the extreme point of median \(m\).
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Hi Bunuel, I am having trouble understanding the m formula. Can you please show me on the image above which of the sides would be considered \(a\)?

Bunuel



• Heron's or Hero's Formula for calculating the area \(A = \sqrt{s(s-a)(s-b)(s-c)}\) where \(a,b,c\) are the three sides of the triangle and \(s = \frac{a+b+c}{2}\) which is the semi perimeter of the triangle.
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Can you explain to me why you would need this formula? I have been trying to find an example of a GMAT question where this formula is needed.


Bunuel

An angle bisector divides the angle into two angles with equal measures.


• Each point of an angle bisector is equidistant from the sides of the angle.
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Can you elaborate on this please? This for some reason isn't making sense to me. I understand an angle bisector is a line divides the angle into two angles with equal measure. What do you mean equidistant from the sides of the angle?

Bunuel

The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment DC is equal to the ratio of the length of side AB to the length of side AC: \(\frac{BD}{DC}=\frac{AB}{AC}\)
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Again I am having a hard time understanding this. Why is it \(\frac{BD}{DC}=\frac{AB}{AC}\) andnot \(\frac{BD}{DC}=\frac{AB}{AD}\)? Thank you for your help
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Math: Triangles 27 Jan 2020, 10:27
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Quote:
Similar Triangles Triangles in which the three angles are identical.

• It is only necessary to determine that two sets of angles are identical in order to conclude that two triangles are similar; the third set will be identical because all of the angles of a triangle always sum to 180º.
• In similar triangles, the sides of the triangles are in some proportion to one another. For example, a triangle with lengths 3, 4, and 5 has the same angle measures as a triangle with lengths 6, 8, and 10. The two triangles are similar, and all of the sides of the larger triangle are twice the size of the corresponding legs on the smaller triangle.
• If two similar triangles have sides in the ratio \(\frac{x}{y}\), then their areas are in the ratio \(\frac{x^2}{y^2}\)


Congruence of triangles Two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in size.

1. SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.

2. SSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.

3. ASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.

So, knowing SAS or ASA is sufficient to determine unknown angles or sides.

NOTE IMPORTANT EXCEPTION:
The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS, or Angle-Side-Side) does not always prove congruence, even when the equal angles are opposite equal sides.

Specifically, SSA does not prove congruence when the angle is acute and the opposite side is shorter than the known adjacent side but longer than the sine of the angle times the adjacent side. This is the ambiguous case. In all other cases with corresponding equalities, SSA proves congruence.

The SSA condition proves congruence if the angle is obtuse or right. In the case of the right angle (also known as the HL (Hypotenuse-Leg) condition or the RHS (Right-angle-Hypotenuse-Side) condition), we can calculate the third side and fall back on SSS.

To establish congruence, it is also necessary to check that the equal angles are opposite equal sides.

So, knowing two sides and non-included angle is NOT sufficient to calculate unknown side and angles.

Angle-Angle-Angle
AAA (Angle-Angle-Angle) says nothing about the size of the two triangles and hence proves only similarity and not congruence.

So, knowing three angles is NOT sufficient to determine lengths of the sides.
Show more

Hi Bunuel! Thanks for this post. Wanted to ask:
What's the difference between similar triangles and congruent triangles? Do they have different properties as well?
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Math: Triangles 27 Jan 2020, 10:34
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Quote:
Similar Triangles Triangles in which the three angles are identical.

• It is only necessary to determine that two sets of angles are identical in order to conclude that two triangles are similar; the third set will be identical because all of the angles of a triangle always sum to 180º.
• In similar triangles, the sides of the triangles are in some proportion to one another. For example, a triangle with lengths 3, 4, and 5 has the same angle measures as a triangle with lengths 6, 8, and 10. The two triangles are similar, and all of the sides of the larger triangle are twice the size of the corresponding legs on the smaller triangle.
• If two similar triangles have sides in the ratio \(\frac{x}{y}\), then their areas are in the ratio \(\frac{x^2}{y^2}\)


Congruence of triangles Two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in size.

1. SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.

2. SSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.

3. ASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.

So, knowing SAS or ASA is sufficient to determine unknown angles or sides.

NOTE IMPORTANT EXCEPTION:
The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS, or Angle-Side-Side) does not always prove congruence, even when the equal angles are opposite equal sides.

Specifically, SSA does not prove congruence when the angle is acute and the opposite side is shorter than the known adjacent side but longer than the sine of the angle times the adjacent side. This is the ambiguous case. In all other cases with corresponding equalities, SSA proves congruence.

The SSA condition proves congruence if the angle is obtuse or right. In the case of the right angle (also known as the HL (Hypotenuse-Leg) condition or the RHS (Right-angle-Hypotenuse-Side) condition), we can calculate the third side and fall back on SSS.

To establish congruence, it is also necessary to check that the equal angles are opposite equal sides.

So, knowing two sides and non-included angle is NOT sufficient to calculate unknown side and angles.

Angle-Angle-Angle
AAA (Angle-Angle-Angle) says nothing about the size of the two triangles and hence proves only similarity and not congruence.

So, knowing three angles is NOT sufficient to determine lengths of the sides.

Hi Bunuel! Thanks for this post. Wanted to ask:
What's the difference between similar triangles and congruent triangles? Do they have different properties as well?
Show more

Similar Triangles are Triangles in which the three angles are identical. Foe example, all equilateral triangles are similar: with side lengths of 1, 1, 1; with side lengths of 2, 2, 2; with side lengths of 3, 3, 3; ... So, you can get a similar to a given triangle by scaling.

Congruent triangles are basically the same triangles. A triangle with side lengths of 3, 4, 5 is congruent only with another triangle with the same side lengths: 3, 4, 5.
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Math: Triangles 27 Jan 2020, 10:42
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Kritisood
Quote:
Similar Triangles Triangles in which the three angles are identical.

• It is only necessary to determine that two sets of angles are identical in order to conclude that two triangles are similar; the third set will be identical because all of the angles of a triangle always sum to 180º.
• In similar triangles, the sides of the triangles are in some proportion to one another. For example, a triangle with lengths 3, 4, and 5 has the same angle measures as a triangle with lengths 6, 8, and 10. The two triangles are similar, and all of the sides of the larger triangle are twice the size of the corresponding legs on the smaller triangle.
• If two similar triangles have sides in the ratio \(\frac{x}{y}\), then their areas are in the ratio \(\frac{x^2}{y^2}\)


Congruence of triangles Two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in size.

1. SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.

2. SSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.

3. ASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.

So, knowing SAS or ASA is sufficient to determine unknown angles or sides.

NOTE IMPORTANT EXCEPTION:
The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS, or Angle-Side-Side) does not always prove congruence, even when the equal angles are opposite equal sides.

Specifically, SSA does not prove congruence when the angle is acute and the opposite side is shorter than the known adjacent side but longer than the sine of the angle times the adjacent side. This is the ambiguous case. In all other cases with corresponding equalities, SSA proves congruence.

The SSA condition proves congruence if the angle is obtuse or right. In the case of the right angle (also known as the HL (Hypotenuse-Leg) condition or the RHS (Right-angle-Hypotenuse-Side) condition), we can calculate the third side and fall back on SSS.

To establish congruence, it is also necessary to check that the equal angles are opposite equal sides.

So, knowing two sides and non-included angle is NOT sufficient to calculate unknown side and angles.

Angle-Angle-Angle
AAA (Angle-Angle-Angle) says nothing about the size of the two triangles and hence proves only similarity and not congruence.

So, knowing three angles is NOT sufficient to determine lengths of the sides.

Hi Bunuel! Thanks for this post. Wanted to ask:
What's the difference between similar triangles and congruent triangles? Do they have different properties as well?

Similar Triangles are Triangles in which the three angles are identical. Foe example, all equilateral triangles are similar: with side lengths of 1, 1, 1; with side lengths of 2, 2, 2; with side lengths of 3, 3, 3; ... So, you can get a similar to a given triangle by scaling.

Congruent triangles are basically the same triangles. A triangle with side lengths of 3, 4, 5 is congruent only with another triangle with the same side lengths: 3, 4, 5.
Show more

In terms of properties, similar triangles have the property that if the sides are in the ratio a:b then areas will be in the ratio a^2:b^2. Any such properties for congruent triangles?
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Math: Triangles 27 Jan 2020, 10:46
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Kritisood

In terms of properties, similar triangles have the property that if the sides are in the ratio a:b then areas will be in the ratio a^2:b^2. Any such properties for congruent triangles?
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Since all congruent triangles are also similar triangles, then all properties of similar triangles apply to congruent triangles.
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All congruent triangles are similar but all similar triangles not necessarily be congruent.



SIMILAR TRIANGLES means the triangles similar in shape but size might differ, the triangles can be seen as smaller or amplified model of other one.





Congruent triangles are the triangles same in shape and size. they over lap each other completely if placed one above the other.




Kritisood
Quote:
Similar Triangles Triangles in which the three angles are identical.

• It is only necessary to determine that two sets of angles are identical in order to conclude that two triangles are similar; the third set will be identical because all of the angles of a triangle always sum to 180º.
• In similar triangles, the sides of the triangles are in some proportion to one another. For example, a triangle with lengths 3, 4, and 5 has the same angle measures as a triangle with lengths 6, 8, and 10. The two triangles are similar, and all of the sides of the larger triangle are twice the size of the corresponding legs on the smaller triangle.
• If two similar triangles have sides in the ratio \(\frac{x}{y}\), then their areas are in the ratio \(\frac{x^2}{y^2}\)


Congruence of triangles Two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in size.

1. SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.

2. SSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.

3. ASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.

So, knowing SAS or ASA is sufficient to determine unknown angles or sides.

NOTE IMPORTANT EXCEPTION:
The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS, or Angle-Side-Side) does not always prove congruence, even when the equal angles are opposite equal sides.

Specifically, SSA does not prove congruence when the angle is acute and the opposite side is shorter than the known adjacent side but longer than the sine of the angle times the adjacent side. This is the ambiguous case. In all other cases with corresponding equalities, SSA proves congruence.

The SSA condition proves congruence if the angle is obtuse or right. In the case of the right angle (also known as the HL (Hypotenuse-Leg) condition or the RHS (Right-angle-Hypotenuse-Side) condition), we can calculate the third side and fall back on SSS.

To establish congruence, it is also necessary to check that the equal angles are opposite equal sides.

So, knowing two sides and non-included angle is NOT sufficient to calculate unknown side and angles.

Angle-Angle-Angle
AAA (Angle-Angle-Angle) says nothing about the size of the two triangles and hence proves only similarity and not congruence.

So, knowing three angles is NOT sufficient to determine lengths of the sides.

Hi Bunuel! Thanks for this post. Wanted to ask:
What's the difference between similar triangles and congruent triangles? Do they have different properties as well?
Show more

Show Spoiler
Attachment:
smilaiar tr.png
smilaiar tr.png [ 3.42 KiB | Viewed 6810 times ]
Attachment:
similar-triangle.png
similar-triangle.png [ 61.89 KiB | Viewed 6791 times ]
Attachment:
xcongruent-triangles.png.pagespeed.ic.wQ5OS9G_zl.png
xcongruent-triangles.png.pagespeed.ic.wQ5OS9G_zl.png [ 57.11 KiB | Viewed 6821 times ]
Attachment:
Angle-angle-side_triangle_congruence.svg.png
Angle-angle-side_triangle_congruence.svg.png [ 70.29 KiB | Viewed 6843 times ]
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Math: Triangles 30 May 2020, 17:16
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1. GMATBusters I thought SAS proves congruence and not just similarity - would you mind elaborating the difference?
2. Bunuel , can we also add AAS to the list of congruent proofs, since the definition provided explicitly calls out just the included line? (ASA)

GMATBusters

All congruent triangles are similar but all similar triangles not necessarily be congruent.



SIMILAR TRIANGLES means the triangles similar in shape but size might differ, the triangles can be seen as smaller or amplified model of other one.





Congruent triangles are the triangles same in shape and size. they over lap each other completely if placed one above the other.




Kritisood
Quote:
Similar Triangles Triangles in which the three angles are identical.

• It is only necessary to determine that two sets of angles are identical in order to conclude that two triangles are similar; the third set will be identical because all of the angles of a triangle always sum to 180º.
• In similar triangles, the sides of the triangles are in some proportion to one another. For example, a triangle with lengths 3, 4, and 5 has the same angle measures as a triangle with lengths 6, 8, and 10. The two triangles are similar, and all of the sides of the larger triangle are twice the size of the corresponding legs on the smaller triangle.
• If two similar triangles have sides in the ratio \(\frac{x}{y}\), then their areas are in the ratio \(\frac{x^2}{y^2}\)


Congruence of triangles Two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in size.

1. SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.

2. SSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.

3. ASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.

So, knowing SAS or ASA is sufficient to determine unknown angles or sides.

NOTE IMPORTANT EXCEPTION:
The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS, or Angle-Side-Side) does not always prove congruence, even when the equal angles are opposite equal sides.

Specifically, SSA does not prove congruence when the angle is acute and the opposite side is shorter than the known adjacent side but longer than the sine of the angle times the adjacent side. This is the ambiguous case. In all other cases with corresponding equalities, SSA proves congruence.

The SSA condition proves congruence if the angle is obtuse or right. In the case of the right angle (also known as the HL (Hypotenuse-Leg) condition or the RHS (Right-angle-Hypotenuse-Side) condition), we can calculate the third side and fall back on SSS.

To establish congruence, it is also necessary to check that the equal angles are opposite equal sides.

So, knowing two sides and non-included angle is NOT sufficient to calculate unknown side and angles.

Angle-Angle-Angle
AAA (Angle-Angle-Angle) says nothing about the size of the two triangles and hence proves only similarity and not congruence.

So, knowing three angles is NOT sufficient to determine lengths of the sides.

Hi Bunuel! Thanks for this post. Wanted to ask:
What's the difference between similar triangles and congruent triangles? Do they have different properties as well?

Show Spoiler
Attachment:
smilaiar tr.png
Attachment:
similar-triangle.png
Attachment:
xcongruent-triangles.png.pagespeed.ic.wQ5OS9G_zl.png
Attachment:
Angle-angle-side_triangle_congruence.svg.png
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Math: Triangles 30 May 2020, 18:12
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1) SAS if corresponding sides are equal and included angle equal , it is case of congruency.
If corresponding sides are proportional and included angle is equal, it is case of similarity. Here S1/S2 = k.
If K =1, case of congruency , K not equal to 1, similarity.

2) AAS or ASA is equivalent as sum of the angles in a triangle is fixed = 180.

I hope it is clear

Happy Learning ?

Lazybum
1. GMATBusters I thought SAS proves congruence and not just similarity - would you mind elaborating the difference?
2. Bunuel , can we also add AAS to the list of congruent proofs, since the definition provided explicitly calls out just the included line? (ASA)

GMATBusters

All congruent triangles are similar but all similar triangles not necessarily be congruent.



SIMILAR TRIANGLES means the triangles similar in shape but size might differ, the triangles can be seen as smaller or amplified model of other one.





Congruent triangles are the triangles same in shape and size. they over lap each other completely if placed one above the other.




Kritisood
Quote:
Similar Triangles Triangles in which the three angles are identical.

• It is only necessary to determine that two sets of angles are identical in order to conclude that two triangles are similar; the third set will be identical because all of the angles of a triangle always sum to 180º.
• In similar triangles, the sides of the triangles are in some proportion to one another. For example, a triangle with lengths 3, 4, and 5 has the same angle measures as a triangle with lengths 6, 8, and 10. The two triangles are similar, and all of the sides of the larger triangle are twice the size of the corresponding legs on the smaller triangle.
• If two similar triangles have sides in the ratio \(\frac{x}{y}\), then their areas are in the ratio \(\frac{x^2}{y^2}\)


Congruence of triangles Two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in size.

1. SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.

2. SSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.

3. ASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.

So, knowing SAS or ASA is sufficient to determine unknown angles or sides.

NOTE IMPORTANT EXCEPTION:
The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS, or Angle-Side-Side) does not always prove congruence, even when the equal angles are opposite equal sides.

Specifically, SSA does not prove congruence when the angle is acute and the opposite side is shorter than the known adjacent side but longer than the sine of the angle times the adjacent side. This is the ambiguous case. In all other cases with corresponding equalities, SSA proves congruence.

The SSA condition proves congruence if the angle is obtuse or right. In the case of the right angle (also known as the HL (Hypotenuse-Leg) condition or the RHS (Right-angle-Hypotenuse-Side) condition), we can calculate the third side and fall back on SSS.

To establish congruence, it is also necessary to check that the equal angles are opposite equal sides.

So, knowing two sides and non-included angle is NOT sufficient to calculate unknown side and angles.

Angle-Angle-Angle
AAA (Angle-Angle-Angle) says nothing about the size of the two triangles and hence proves only similarity and not congruence.

So, knowing three angles is NOT sufficient to determine lengths of the sides.

Hi Bunuel! Thanks for this post. Wanted to ask:
What's the difference between similar triangles and congruent triangles? Do they have different properties as well?

Show Spoiler
Attachment:
smilaiar tr.png
Attachment:
similar-triangle.png
Attachment:
xcongruent-triangles.png.pagespeed.ic.wQ5OS9G_zl.png
Attachment:
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Math: Triangles 14 Jun 2023, 21:30
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Hi Bunuel,

Thank you for such an amazing job. I have a couple of questions:

1) Is this theory about triangle enough to solve any triangle problem on GMAT?

2)Are the following concepts are tested on GMAT?
-centroid of the triangle
-Other area formulas :
• A=P∗r/2
• A=abc/4R
-midsegment of a triangle
- congurence of triangles

Thanks

1. Should be enough.

2. Knowing more would not hurt. I've seen questions involving congruence and mid-segment concepts. Cannot recall a geometry problem which could have been solved ONLY by some lesser know formulae.

23. Geometry




24. Coordinate Geometry




25. Triangles




26. Polygons




27. Circles




28. Rectangular Solids and Cylinders




29. Graphs and Illustrations



For more check Ultimate GMAT Quantitative Megathread
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Bunuel This is a wonderful post! Thank you. Do you have similar posts on other Quant concepts?
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Math: Triangles 14 Jun 2023, 23:47
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Bunuel This is a wonderful post! Thank you. Do you have similar posts on other Quant concepts?
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For more:
Ultimate GMAT Quantitative Megathread
ALL YOU NEED FOR QUANT ! ! !

Hope it helps.
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Tuck School Moderator
853 posts