Similar triangles
45 flashcards covering Similar triangles for the SAT Math section.
Similar triangles are triangles that have the same shape but not necessarily the same size. This means their corresponding angles are equal, and their corresponding sides are proportional, creating a consistent ratio between them. For example, if you enlarge a photo of a triangle, the new triangle will be similar to the original because the angles stay the same and the sides scale up evenly. This concept is fundamental in geometry for solving problems involving proportions and scaling.
On the SAT Math section, similar triangles show up in questions that ask you to find missing side lengths, compare figures, or apply ratios in real-world scenarios like shadows or maps. Common traps include mistaking similar triangles for congruent ones or setting up proportions incorrectly, which can lead to wrong answers. Focus on identifying key features like equal angles and proportional sides, and practice solving for unknowns quickly, as these problems often combine with other topics like algebra.
A helpful tip: Always double-check that your triangles meet both angle and side ratio criteria before proceeding.
Terms (45)
- 01
Similar Triangles
Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional, meaning they have the same shape but not necessarily the same size.
- 02
AA Similarity Postulate
If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.
- 03
SAS Similarity Theorem
If two sides of one triangle are proportional to two sides of another triangle and the included angles are equal, then the triangles are similar.
- 04
SSS Similarity Theorem
If the corresponding sides of two triangles are proportional, then the triangles are similar.
- 05
Corresponding Angles
In similar triangles, angles that are in the same relative position are equal, such as the angle opposite the longest side in each triangle.
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Corresponding Sides
In similar triangles, sides that are opposite corresponding angles are proportional, meaning their lengths form the same ratio.
- 07
Scale Factor
The ratio by which all corresponding linear measurements of similar triangles are multiplied to get from one triangle to the other, such as 2 if one triangle is twice as large.
- 08
Proportional Sides
In similar triangles, the ratios of the lengths of corresponding sides are equal, allowing you to set up equations like a/b = c/d.
- 09
Setting Up Proportions
To solve problems with similar triangles, write a proportion equating the ratios of corresponding sides, ensuring the order matches the triangles' correspondence.
- 10
Solving for Unknown Sides
Use the proportion from similar triangles to solve for a missing side length by cross-multiplying and simplifying the equation.
- 11
Indirect Measurement
Similar triangles help measure inaccessible heights or distances by comparing them to a known similar object, like using a mirror to find a building's height.
- 12
Shadow Problems
In problems involving shadows, triangles formed by objects and their shadows are similar, allowing you to set up proportions to find unknown heights based on known lengths.
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Height of Objects
Use similar triangles to calculate the height of an object by comparing it to a smaller similar triangle with a known height, such as a person standing next to a flagpole.
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Similar Right Triangles
Right triangles are similar if they have an acute angle in common, since the right angles are also equal, making AA similarity apply.
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Geometric Mean
In similar right triangles, such as those formed by an altitude to the hypotenuse, the altitude is the geometric mean of the two segments it creates on the hypotenuse.
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Altitude to Hypotenuse
In a right triangle, the altitude drawn to the hypotenuse creates two smaller triangles similar to each other and to the original triangle.
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Trap: Assuming Similarity
A common error is assuming triangles are similar just because they look alike; always verify with AA, SAS, or SSS criteria.
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Congruent vs. Similar Triangles
Congruent triangles are identical in shape and size, while similar triangles are identical in shape but not necessarily size, so check for equal sides if needed.
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Ratio of Perimeters
For similar triangles, the ratio of their perimeters equals the scale factor, meaning if sides are in a 2:1 ratio, perimeters are too.
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Ratio of Areas
For similar triangles, the ratio of their areas is the square of the scale factor, so if sides are in a 2:1 ratio, areas are in a 4:1 ratio.
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Dilations
A dilation is a transformation that enlarges or reduces a figure by a scale factor from a fixed point, resulting in similar triangles.
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Center of Dilation
The fixed point from which a dilation occurs, and in similar triangles, lines from this center to corresponding vertices are proportional.
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Enlarging Triangles
When a triangle is enlarged by a scale factor greater than 1, it creates a similar triangle with all sides multiplied by that factor.
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Reducing Triangles
When a triangle is reduced by a scale factor less than 1, it creates a similar triangle with all sides divided by the reciprocal of that factor.
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Parallel Lines and Similar Triangles
If a line parallel to one side of a triangle intersects the other two sides, it creates a smaller triangle similar to the original.
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Transversals Creating Similar Triangles
When a transversal crosses parallel lines, it can form similar triangles by creating equal corresponding angles.
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Pythagorean Theorem in Similar Triangles
In similar right triangles, the Pythagorean theorem applies to each, and you can use side ratios to relate their hypotenuses and legs.
- 28
Trigonometry Ratios in Similar Triangles
In similar triangles, sine, cosine, and tangent ratios for corresponding angles are equal, allowing you to transfer ratios between triangles.
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Example: Two Triangles with Equal Angles
If one triangle has angles of 30°, 60°, and 90°, and another has the same, they are similar, with sides proportional based on their scale.
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Example: Triangles with Sides in Proportion
If triangle ABC has sides 3, 4, 5 and triangle DEF has sides 6, 8, 10, they are similar because the sides are in a 2:1 ratio.
- 31
Strategy for Identifying Similar Triangles
First, check for equal angles or proportional sides; if neither is obvious, redraw the diagram to align corresponding parts.
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Common Proportion Setup
When setting up a proportion for similar triangles, ensure the ratios compare corresponding sides, like shorter to shorter and longer to longer.
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Using Cross-Multiplication
After writing a proportion from similar triangles, cross-multiply to solve for unknowns, but double-check for consistent units.
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Trap: Scale Factor Misinterpretation
Remember that the scale factor is the ratio of a side in the larger triangle to the corresponding side in the smaller one, not vice versa.
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Area Scale in Word Problems
In problems involving similar triangles and areas, square the scale factor to find the area ratio, such as for comparing painted surfaces.
- 36
Perimeter in Similar Figures
For similar triangles, calculate perimeter by multiplying the perimeter of one by the scale factor, useful in fencing or boundary problems.
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Mirror Reflection Problems
Similar triangles form when using a mirror for indirect measurement, with the angle of incidence equaling the angle of reflection.
- 38
Slope and Similar Triangles
In coordinate geometry, triangles with the same slope for corresponding sides may be similar if angles match, helping in graphing problems.
- 39
Enlargement from a Point
Similar triangles can be created by drawing lines from a point to the vertices and extending them, with distances scaled accordingly.
- 40
Reduction in Diagrams
In scaled diagrams, smaller triangles are similar to larger ones, and you can use the scale to find actual measurements.
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Trap: Non-Corresponding Sides
Avoid mixing up which sides correspond; always label triangles to match angles before setting up ratios.
- 42
Proportions with Variables
In equations involving similar triangles, solve for variables in proportions by isolating them, such as x/5 = 4/10.
- 43
Real-World Scale Models
Similar triangles represent scale models, like architectural plans, where measurements are proportional to the actual structure.
- 44
Angle Bisector Theorem
An angle bisector in a triangle divides the opposite side in the ratio of the adjacent sides, creating similar triangles.
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Midsegment Theorem
A midsegment of a triangle is parallel to the third side and half as long, forming a smaller similar triangle.