Plane geometry triangles
53 flashcards covering Plane geometry triangles for the GMAT Quantitative section.
Plane geometry triangles are two-dimensional shapes formed by three straight lines connecting at three points, creating a closed figure. They are fundamental in geometry and appear in everyday applications like engineering and design. Key aspects include the types of triangles—equilateral, isosceles, scalene, and right-angled—as well as properties such as the sum of interior angles equaling 180 degrees, the Pythagorean theorem for right triangles, and formulas for area (like 1/2 base times height) and perimeter.
On the GMAT Quantitative section, triangles show up in problem-solving and data sufficiency questions, often requiring you to calculate areas, angles, or side lengths, or to identify similarities and congruences. Common traps include misapplying the triangle inequality theorem, overlooking special properties like the 30-60-90 triangle ratios, or making errors in diagram interpretation. Focus on understanding core theorems, practicing quick sketches, and verifying assumptions to handle these efficiently.
A concrete tip: Always draw a triangle to scale when solving problems.
Terms (53)
- 01
Equilateral triangle
A triangle where all three sides are equal in length, and all three angles are equal to 60 degrees.
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Isosceles triangle
A triangle with at least two sides of equal length, resulting in at least two equal angles opposite those sides.
- 03
Scalene triangle
A triangle where all three sides have different lengths, and all three angles are of different measures.
- 04
Acute triangle
A triangle where all three interior angles are less than 90 degrees.
- 05
Obtuse triangle
A triangle with one interior angle greater than 90 degrees and the other two less than 90 degrees.
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Right triangle
A triangle with one interior angle exactly equal to 90 degrees.
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Sum of angles in a triangle
The total measure of the three interior angles of any triangle is always 180 degrees.
- 08
Exterior angle of a triangle
An angle formed by extending one side of a triangle, which is equal to the sum of the two non-adjacent interior angles.
- 09
Pythagorean theorem
In a right triangle, the square of the hypotenuse's length equals the sum of the squares of the other two sides.
- 10
Area of a triangle
The measure of the space inside a triangle, calculated as one-half times base times height.
- 11
Perimeter of a triangle
The total length around a triangle, found by adding the lengths of all three sides.
- 12
Triangle inequality theorem
In any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
- 13
Congruent triangles
Two triangles that have exactly the same size and shape, with all corresponding sides and angles equal.
- 14
Similar triangles
Two triangles that have the same shape but not necessarily the same size, with corresponding angles equal and corresponding sides proportional.
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SSS congruence
A rule stating that two triangles are congruent if all three pairs of corresponding sides are equal in length.
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SAS congruence
A rule stating that two triangles are congruent if two pairs of corresponding sides and the included angle are equal.
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ASA congruence
A rule stating that two triangles are congruent if two pairs of corresponding angles and the included side are equal.
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AAS congruence
A rule stating that two triangles are congruent if two pairs of corresponding angles and a non-included side are equal.
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HL congruence
A rule for right triangles stating that they are congruent if the hypotenuse and one leg of one triangle are equal to the hypotenuse and one leg of the other.
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AA similarity
A rule stating that two triangles are similar if two pairs of corresponding angles are equal.
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triangle
A special right triangle with angles of 30 degrees, 60 degrees, and 90 degrees, where the sides are in the ratio 1 : √3 : 2.
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Median of a triangle
A line segment joining a vertex to the midpoint of the opposite side, dividing the triangle into two triangles of equal area.
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Altitude of a triangle
A perpendicular line segment from a vertex to the line containing the opposite side, representing the height for area calculations.
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Angle bisector
A line that divides an angle of a triangle into two equal angles, extending from the vertex to the opposite side.
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Perpendicular bisector
A line that cuts a side of a triangle into two equal parts at a right angle, passing through the midpoint.
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Centroid of a triangle
The point where the three medians intersect, serving as the center of mass and dividing each median into a 2:1 ratio.
- 27
Orthocenter of a triangle
The point where the three altitudes intersect, which can be inside or outside the triangle depending on its type.
- 28
Circumcenter of a triangle
The center of the circle that passes through all three vertices, located at the intersection of the perpendicular bisectors.
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Incenter of a triangle
The center of the circle tangent to all three sides, found at the intersection of the angle bisectors.
- 30
Heron's formula
A method to calculate the area of a triangle when all three side lengths are known, using the square root of [s(s-a)(s-b)(s-c)], where s is the semi-perimeter.
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Base of a triangle
Any one of the sides of a triangle, typically the bottom side, used as the base in area formulas with the corresponding height.
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Height of a triangle
The perpendicular distance from the base to the opposite vertex, essential for calculating the area.
- 33
Law of sines
In any triangle, the ratio of the length of a side to the sine of its opposite angle is constant for all three sides.
- 34
Law of cosines
A formula relating the lengths of the sides of a triangle to the cosine of one of its angles, useful for non-right triangles.
- 35
Common trap: Assuming congruence
A frequent error where triangles are assumed congruent based on only one or two matching sides or angles, ignoring required criteria.
- 36
Strategy for similar triangles
To solve problems, identify corresponding angles and set up proportions between corresponding sides to find unknown lengths.
- 37
Distance formula in triangles
Use the formula √[(x2-x1)² + (y2-y1)²] to find the length of a side when triangle vertices are given as coordinates.
- 38
Midpoint theorem
In a triangle, the segment joining the midpoints of two sides is parallel to the third side and half as long.
- 39
Example: Area of right triangle
For a right triangle with legs 3 and 4, the area is (1/2) 3 4 = 6.
This illustrates the basic area formula for a right triangle.
- 40
Example: Pythagorean triple
A set like 5, 12, 13 forms a Pythagorean triple because 5² + 12² = 25 + 144 = 169 = 13².
Useful for recognizing common right triangles quickly.
- 41
Example: Similar triangles ratio
If two triangles are similar with a side ratio of 2:3, then all corresponding sides maintain that ratio.
A side of 4 in the first triangle corresponds to 6 in the second.
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Common trap: Angle sum error
Mistakenly adding angles in a triangle to more or less than 180 degrees, often due to misidentifying exterior angles.
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Strategy for triangle inequality
Check if three lengths can form a triangle by verifying that the sum of any two exceeds the third, to avoid invalid configurations.
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Equilateral triangle properties
All sides equal and all angles 60 degrees, making it equilateral, equiangular, and with equal altitudes, medians, and angle bisectors.
- 45
Isosceles triangle properties
The base angles are equal, and the altitude from the apex to the base is also the median and angle bisector.
- 46
Scalene triangle properties
No sides or angles are equal, requiring individual measurement for all elements in problems.
- 47
Right triangle properties
One angle is 90 degrees, and the sides satisfy the Pythagorean theorem, with the hypotenuse as the longest side.
- 48
Example: Heron's formula application
For a triangle with sides 5, 6, and 7, the semi-perimeter is 9, and area is √[9(9-5)(9-6)(9-7)] = √[9432] = √216 ≈ 14.7.
This shows how to use it when height is unknown.
- 49
Common trap: Similar vs congruent
Confusing similar triangles, which only require proportional sides, with congruent ones that must be identical in size.
- 50
Strategy for angle bisectors
Use angle bisectors to divide angles equally and find points like the incenter, which is equidistant from all sides.
- 51
Obtuse triangle area calculation
Even in an obtuse triangle, area is one-half base times height, where height is the perpendicular from the opposite vertex.
- 52
Acute triangle circumcircle
In an acute triangle, the circumcenter lies inside the triangle, as the perpendicular bisectors intersect within.
- 53
Example: SAS similarity
If two triangles have two pairs of sides proportional and the included angles equal, they are similar.
Triangles with sides 2,3 and included angle 40°, and 4,6 with the same angle.