Transformations
Congruence & Proof
Similarity & Dilations
Right Triangles
Coordinate Geometry
100
A triangle is reflected across a line and then translated. Do these rigid motions preserve distance and angle measure? Explain briefly.
Example Response: Yes — rigid motions (translations, rotations, reflections) preserve distance and angle measure, so congruence is preserved.
100
What does it mean for two figures to be congruent?
Congruent figures have the same shape and size; corresponding sides and angles are equal.
100
What does it mean for polygons to be similar?
Similar polygons have the same shape, but not necessarily the same size; corresponding angles are congruent and corresponding sides are proportional.
100
State the Pythagorean Theorem
a2+b2=c2 (or that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse)
100
A line segment has endpoints A(2,1) and B(5,3). What is the slope AB in simplest form?
2/3
200
Describe how the coordinates of a point (x,y) change under the transformation rule (x,y)→(x+5,y+2). (Also give the name of the transformation)
Translation right 5 and up 2
200
List three rigid motions that preserve congruence.
Translations, rotations, and reflections.
200
A dilation with center at the origin and scale factor of 2 sends point (3,4) to what image?
(3,4)→(6,8)
200
In a right triangle with legs of lengths 7 and 24, find the hypotenuse.
Hypotenuse = 25
200
Find the distance between points (−3,4) and (5,−2)
10
300
Give the transformation rule for a reflection across the x-axis on a point (a,b).
(a,b)→(a,-b)
300
Given triangles △ABC and △DEF with AB=DE, ∠B=∠E, and BC=EF, name an appropriate congruence theorem (pattern) that could be used to prove △ABC≅△DEF.
SAS
300
If two triangles are similar with scale factor 3/5 from triangle P to triangle Q, and a side in triangle P measures 20 cm, what is the corresponding side length in triangle Q?
20*3/5 = 12 cm
300
In a right triangle, give the definition for the sine, cosine, and tangent ratios (in terms of the sides).
sinθ = hypotenuse/opposite, cosθ = hypotenuse/adjacent, tanθ = adjacent/opposite (SOH-CAH-TOA!)
300
What is the midpoint of the points (3,-1) and (5,3)?
(4,1)
400
What is the image of (x,y) under a rotation of 90° counterclockwise about the origin?
(-y,x)
400
Provide two examples of reasons commonly used in a two-column geometric proof.
Reasons column includes definitions, given information, postulates/theorems (e.g., Vertical Angles Theorem, SAS, CPCTC).
400
Given that △ABC ~ △DEF, where the scale factor from △ABC to △DEF is 3, if ∠B=50°, what is the measure of ∠E?
∠E=50°
400
A right triangle has an acute angle θ where tanθ = 3/4. Find sinθ in simplest form.
sinθ = 3/5
400
Find the equation of the line in slope-intercept form that passes through (2,3) with slope −2.
y=−2x+7
500
Given the composition: rotate 180° about the origin, then reflect across the line y=x. Find the resulting mapping of point (x,y).
(−y,−x)
500
What is the missing reason?
Vertical angles are congruent
500
Given two similar triangles, △ABC ~ △DEF, where AC=10, AB=6, and DE=15, what is DF?
DF=25
500
A ladder leans against a wall making an angle of θ with the ground; the top reaches a height of 15 feet and the base is 9 feet from the wall. Determine the angle the ladder makes with the ground (calculate to nearest degree)
59°
500
Determine whether the quadrilateral ABCD with vertices A(0,0), B(4,0), C(5,3), and D(1,3) is a parallelogram. Explain your reasoning using slopes or side lengths.
Opposite sides have equal slopes (AB and CD have slope 0 and BC and DA have slope 3) and equal side lengths (AB=CD=4, and BC=DA=√10), so it is a parallelogram.