Triangles
58 flashcards covering Triangles for the SAT Math section.
Triangles are fundamental shapes in geometry, defined as a polygon with three sides and three angles that sum to 180 degrees. They come in various types, such as equilateral, isosceles, scalene, and right triangles, each with unique properties. Understanding triangles helps build skills in spatial reasoning and problem-solving, which are essential for more advanced math topics.
On the SAT Math section, triangles show up in questions about area, perimeter, the Pythagorean theorem, similarity, and congruence. You'll encounter multiple-choice problems that test your ability to apply formulas or identify relationships, with common traps like misapplying the triangle inequality or confusing acute versus obtuse angles. Focus on mastering key theorems and visualizing diagrams to avoid errors in word problems or coordinate geometry.
Always double-check your angle sums in any triangle problem.
Terms (58)
- 01
Equilateral Triangle
A triangle where all three sides are equal in length, and all three angles are equal to 60 degrees.
- 02
Isosceles Triangle
A triangle with at least two sides of equal length, and the angles opposite those sides are also equal.
- 03
Scalene Triangle
A triangle where all three sides have different lengths, and all three angles are different.
- 04
Right Triangle
A triangle with one angle measuring exactly 90 degrees, often used with the Pythagorean theorem to relate the sides.
- 05
Acute Triangle
A triangle where all three interior angles are less than 90 degrees.
- 06
Obtuse Triangle
A triangle with one interior angle greater than 90 degrees and the other two less than 90 degrees.
- 07
Triangle Sum Theorem
The rule that the sum of the interior angles in any triangle is always 180 degrees.
- 08
Pythagorean Theorem
A formula for right triangles stating that the square of the hypotenuse's length equals the sum of the squares of the other two sides, or a² + b² = c².
- 09
Area of a Triangle
The measure of the space inside a triangle, calculated as one-half times base times height.
- 10
Perimeter of a Triangle
The total length around a triangle, found by adding the lengths of all three sides.
- 11
Congruent Triangles
Two triangles that are identical in shape and size, with all corresponding sides and angles equal.
- 12
Similar Triangles
Two triangles that have the same shape but not necessarily the same size, with corresponding angles equal and corresponding sides proportional.
- 13
SSS Congruence
A criterion for triangle congruence where if three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
- 14
SAS Congruence
A criterion for triangle congruence where if two sides and the included angle of one triangle are equal to two sides and the included angle of another, the triangles are congruent.
- 15
ASA Congruence
A criterion for triangle congruence where if two angles and the included side of one triangle are equal to two angles and the included side of another, the triangles are congruent.
- 16
AAS Congruence
A criterion for triangle congruence where if two angles and a non-included side of one triangle are equal to two angles and the corresponding non-included side of another, the triangles are congruent.
- 17
HL Congruence
A criterion for right triangle congruence where if the hypotenuse and one leg of one right triangle are equal to the hypotenuse and one leg of another, the triangles are congruent.
- 18
AA Similarity
A criterion for triangle similarity where if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
- 19
SSS Similarity
A criterion for triangle similarity where if the corresponding sides of two triangles are proportional, the triangles are similar.
- 20
SAS Similarity
A criterion for triangle similarity where if two sides of one triangle are proportional to two sides of another and the included angles are equal, the triangles are similar.
- 21
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.
- 22
Median of a Triangle
A line segment joining a vertex of a triangle to the midpoint of the opposite side, used to divide the triangle into two equal-area triangles.
- 23
Altitude of a Triangle
A perpendicular line segment from a vertex to the line containing the opposite side, representing the height for area calculations.
- 24
Angle Bisector
A line that divides an angle of a triangle into two equal angles, and it divides the opposite side in the ratio of the adjacent sides.
- 25
Perpendicular Bisector
A line that cuts a side of a triangle into two equal parts at a right angle, passing through the midpoint.
- 26
Exterior Angle Theorem
The theorem stating that an exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
- 27
Triangle Inequality Theorem
The rule that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
- 28
Heron's Formula
A formula to find the area of a triangle when all three side lengths are known, calculated as √[s(s-a)(s-b)(s-c)], where s is the semi-perimeter.
- 29
Sine in a Right Triangle
The ratio of the length of the opposite side to the hypotenuse in a right triangle, used to find missing sides or angles.
- 30
Cosine in a Right Triangle
The ratio of the length of the adjacent side to the hypotenuse in a right triangle, helpful for solving for unknowns.
- 31
Tangent in a Right Triangle
The ratio of the length of the opposite side to the adjacent side in a right triangle, often used in angle-related problems.
- 32
SOH-CAH-TOA
A mnemonic for trigonometric ratios: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.
- 33
Common Trap: Assuming Congruence
A frequent error where students assume triangles are congruent without verifying all necessary criteria, such as SSS or SAS.
- 34
Common Trap: Angle Misidentification
A mistake where students confuse acute and obtuse angles in triangles, leading to incorrect applications of theorems.
- 35
Distance Formula in Triangles
A method using the formula √[(x2-x1)² + (y2-y1)²] to find side lengths of a triangle in the coordinate plane.
- 36
Midpoint Formula for Triangles
A formula to find the midpoint of a side in a triangle, calculated as ((x1+x2)/2, (y1+y2)/2), useful for medians.
- 37
Slope in Triangle Problems
The measure of a line's steepness, calculated as rise over run, used to determine if sides of a triangle are perpendicular.
- 38
Example: Pythagorean Application
In a right triangle with legs of 3 and 4, the hypotenuse is 5, as 3² + 4² = 9 + 16 = 25, and √25 = 5.
For sides 3 and 4, hypotenuse is 5.
- 39
Example: Area Calculation
For a triangle with base 6 and height 4, the area is (1/2) 6 4 = 12 square units.
- 40
Example: Similar Triangles
If two triangles have angles 30°, 60°, 90° and sides in ratio 1:2, the smaller has sides half as long as the larger.
- 41
Example: Triangle Inequality
For sides 2, 3, and 4, it works since 2+3>4, 2+4>3, and 3+4>2, but 1, 2, 4 fails because 1+2=3 is not greater than 4.
- 42
Example: 30-60-90 Sides
In a 30-60-90 triangle with shortest side 5, the other leg is 5√3 and hypotenuse is 10.
- 43
Example: ASA Congruence
If two triangles each have angles 40° and 60° with the included side 5 units, they are congruent.
- 44
Example: Sine Calculation
In a right triangle with opposite side 3 and hypotenuse 5, sine of the angle is 3/5 or 0.6.
- 45
Example: Exterior Angle
In a triangle with angles 50° and 60°, the exterior angle at the third vertex is 50° + 60° = 110°.
- 46
Example: Heron's Formula Use
For a triangle with sides 5, 6, and 7, semi-perimeter is 9, so area is √[9(9-5)(9-6)(9-7)] = √[9432] ≈ 8.66.
- 47
Centroid of a Triangle
The point where the three medians intersect, serving as the center of mass and dividing each median in a 2:1 ratio.
- 48
Orthocenter of a Triangle
The point where the three altitudes intersect, located inside for acute triangles and outside for obtuse ones.
- 49
Circumcenter of a Triangle
The center of the circle passing through all three vertices, found at the intersection of perpendicular bisectors.
- 50
Incenter of a Triangle
The center of the incircle, where the angle bisectors meet, and it's equidistant from all sides.
- 51
Common Trap: Scale Factor
An error in similar triangles where students forget that all sides scale proportionally, affecting area by the square of the factor.
- 52
Common Trap: Right Triangle Angles
Mistakenly assuming the largest angle is always opposite the longest side, which is true but often overlooked in non-right triangles.
- 53
Example: Tangent Calculation
In a right triangle with opposite side 4 and adjacent side 3, tangent of the angle is 4/3 or approximately 1.333.
- 54
Example: Perimeter Calculation
For a triangle with sides 7, 8, and 9, the perimeter is 7 + 8 + 9 = 24 units.
- 55
Example: Angle Bisector Theorem
In a triangle with sides 5 and 7 adjacent to the bisected angle, the bisector divides the opposite side in the ratio 5:7.
- 56
Example: Coordinate Triangle Area
For vertices at (0,0), (3,0), and (0,4), the area is (1/2)| (00 + 34 + 00) - (03 + 00 + 40) | = (1/2)|12| = 6.
- 57
Common Trap: Median vs. Altitude
Confusing a median, which goes to the midpoint, with an altitude, which is perpendicular, leading to incorrect geometric constructions.
- 58
Example: SAS Similarity
If two triangles have sides 2 and 3 with included angle 40°, and another has sides 4 and 6 with included angle 40°, they are similar.