Hidden Similarity
When an altitude is drawn to the hypotenuse, how many triangles are created?
3
The altitude to the hypotenuse is the geometric mean of what two values?
The two segments of the hypotenuse
What ratio compares opposite to hypotenuse?
Sine
A ladder problem forms what type of triangle?
Right triangle
What does slope measure?
Steepness (rise/run)
If a triangle is scaled by 2, how does area change?
×4
What makes all three triangles similar?
AA similarity
Each leg is the geometric mean between what two lengths?
Hypotenuse and its projection
If sin(θ) = 1/2, what is θ?
30°
If a rope swings like a pendulum, what geometric shape is formed by its path?
Arc of a circle
If slope is negative, what does the graph do?
Decreases left to right
If tan(θ) = 3/4, find sin(θ).
3/5
If two triangles share an acute angle and both are right, what can you conclude?
They are similar
If hypotenuse is 25 and one projection is 9, what is the corresponding leg?
15
If tan(θ) = 2, what does that tell you about the triangle?
Opposite is twice adjacent
Why does a long object slightly bending create an isosceles triangle model?
Ends stay fixed while middle shifts
What is needed to write a line equation?
Slope and intercept
A triangle has sides 10, 24, 26. What type is it?
Right triangle
Why does the altitude create two smaller triangles similar to the original?
Shared angles + right angles
If the hypotenuse is split into 4 and 9, what is the altitude?
6
A point lies on a circle of radius 1. What are its coordinates based on angle θ?
(cosθ, sinθ)
If two angles from a shoreline are known, what math idea helps find distance?
Triangulation (trig/similarity)
Where does a graph cross the x-axis?
Where y = 0
If altitude = √(xy), what must x and y represent?
Segments of hypotenuse
If two triangles are each similar to a third triangle, what must be true?
They are similar to each other
Why do geometric mean relationships exist in right triangles?
Because of similarity between triangles
Why do trig ratios stay constant for the same angle?
Because all such triangles are similar
Why are right triangles useful in real-world measurement?
They allow indirect measurement using ratios
Why can motion paths be modeled with linear equations?
Constant rate of change
Explain why trig and similarity are connected.
Trig ratios come from similar triangles