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R13 - Triangles (2/2)
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What do you know about the height of a triangle with an obtuse angle?
For triangles with obtuse angles, the height of the triangle lies outside of the triangle.
Similarity of triangles
- Rule
- What do SIMILAR triangles have in common?
- Implications for area & volume
Triangles are similar, if they have the same shape but not necessarily the same sizes.
Characteristics: (1) Corresponding angles are the same in measure and (2) corresponding sides (incl. height) are the same proportion
If two similar triangles have corresponding side lengths in ratio a:b, then their areas will be in ratio a^2:b^2. For similar solids, their volumes will be on ratio a^3:b^3.
The principle holds for any similar figures (other polygons).
What are the 3 standard ways to identify SIMILAR triangles? (Triggers)
1) match at least two sets of angles (RIGHT triangles: only one other angle needs to be same)
2) constant ratio of corresponding sides (all 3 sides)
3) constant ratio for 2 sets of SIDES -and- if the single ANGLE measurement between them is the same
TIP: Look out for similarity, if one triangle is part of the other.
Similarity of triangles (technique)
(1) Introduce a variable that expresses proportionality (e.g. k)
(2) In order to allocate prportions correctly, turn the triangle mentally so that the right angle + hypothenuse match
Ex.: The area of the larger triangle (A1) is twice as large as the area of the smaller triangle (A2). What is the side length of the smaller in terms of the side length of the larger triangle? (triangles are similiar)
A1 = 2A2
k = constant (expressing ratio)
A1 in terms of A2: A1 = (2)(1/2)(s)(h) = (1/2)(k)(s)(k)(h)
sh = (k^2(s)(h))/2
k = √(2)
Thus, S = s*√(2)
What does "Two triangles are congruent" mean?
Corresponding sides and angles are equal in size.
Congruence of triangles (triggers & traps)
1) SAS (side-angle-side) are same in both triangles
2) SSS (side-side-side) are same in both triangles
3) ASA (angle-side-angle) are same in both triangles
--> if these parts of two triangles correspond, then this is sufficient to determine all remaining unknown sides and angles.
TRAPS:
A) SSA is insufficient (angle must be included between given sides)
B) AAA is insufficient (just knowing all three angles, does not tell you anything about the size fo the triangle)
what is the formula for the AREA of any triangle?
1/2 (base * height)
Note: The height needs to be perpendicular to the base (shorter legs in right triangle
If 7 and 10 are the lengths of two sides of a triangular region, which of the following can be the length of the third side?
a) 2 b) 8 c) 17
b) 8 (greater than 10-7 = 3 and less than 7+10 = 17, i.e. the number has to be in the range between 4 to 16).
what do GMAT pimps know when they see the term 'not drawn to scale' next to figures?
these figures are intentionally manipulated to sucker you into making false assumptions. "trust the information, not the representation."
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