Dilations
This type of transformation changes the size of a figure but preserves its shape and angle measures.
What is a dilation?
Two triangles are similar if their corresponding angles are congruent and their corresponding sides are in this relationship.
What is proportional (or in the same ratio)?
This triangle similarity theorem states that if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
What is AA (Angle-Angle) Similarity?
This theorem states that if a line is parallel to one side of a triangle, it divides the other two sides proportionally.
What is the Triangle Proportionality Theorem (or Basic Proportionality Theorem)?
In a right triangle, this theorem states that a² + b² = c², where c is the hypotenuse.
What is the Pythagorean Theorem?
When a triangle is dilated with scale factor 3, each side length is multiplied by this number.
What is 3?
Triangle ABC has sides 6, 8, 10. Triangle DEF has sides 9, 12, 15. This is the scale factor from ABC to DEF.
What is 3:2 or 1.5?
Triangle ABC has angles 45°, 75°, and 60°. Triangle DEF has angles 45° and 75°. This is the measure of the third angle in triangle DEF.
What is 60°? (The triangles are similar by AA)
In triangle ABC, DE is parallel to BC. If AD = 6, DB = 4, and AE = 9, then EC equals this.
What is 6? (By proportionality: 6/4 = 9/EC, so EC = 6)
A right triangle has legs of length 5 and 12. This is the length of the hypotenuse.
What is 13? (5² + 12² = 25 + 144 = 169 = 13²)
Point A(4, 6) is dilated by scale factor 1/2 with center at origin. These are the coordinates of A'.
What is (2, 3)?
Two similar triangles have a scale factor of 2:3. If the smaller triangle has area 12 square units, the larger triangle has this area.
What is 27 square units? (Areas scale by the square of the ratio: 12 × (3/2)² = 12 × 9/4 = 27)
This similarity theorem requires two pairs of proportional sides and the included angle to be congruent.
What is SAS (Side-Angle-Side) Similarity?
In triangle PQR, ST is parallel to QR. PS = 8, SQ = 12, PT = 6. This is the length of TR.
What is 9? (8/12 = 6/TR, so TR = 9)
In a right triangle, when you draw the altitude to the hypotenuse, it creates two smaller triangles that have this relationship to the original triangle and to each other.
What is similar? (All three triangles are similar to each other)
A rectangle has perimeter 20. After dilation with scale factor 2.5, the new perimeter is this.
What is 50?
A flagpole casts a 15-foot shadow while a 6-foot person casts a 4-foot shadow. This is the height of the flagpole.
What is 22.5 feet? (6/4 = h/15, so h = 22.5)
Triangle ABC has sides AB = 8, BC = 12, CA = 16. Triangle DEF has sides DE = 6, EF = 9, FD = 12. These triangles are similar by this theorem.
What is SSS (Side-Side-Side) Similarity? (All sides are in ratio 3:4)
A triangle has sides of length 15, 20, and 25. A line parallel to the longest side creates a smaller triangle with perimeter 24. This is the length of the side parallel to the 25-unit side.
What is 10? (Scale factor is 24/60 = 2/5, so parallel side = 25 × 2/5 = 10)
In a 30-60-90 triangle, if the shortest side is 6, then the hypotenuse has this length.
What is 12? (In 30-60-90 triangles, sides are in ratio 1:√3:2)
Under dilation, this property is preserved: angle measures, parallel lines, or distances between points.
What are angle measures and parallel lines? (Note: distances change by the scale factor)
Two similar triangles have perimeters of 24 and 36. If the area of the smaller triangle is 32, this is the area of the larger triangle.
What is 72? (Scale factor is 36/24 = 3/2, area scales by (3/2)² = 9/4, so 32 × 9/4 = 72)
In triangle ABC, angle A = 53°, AB = 10, AC = 15. In triangle DEF, angle D = 53°, DE = 6, DF = 9. This theorem proves the triangles are similar.
What is SAS Similarity? (Angle A = Angle D, and sides are proportional 10/6 = 15/9 = 5/3)
Triangle ABC has a cevian AD where D is on BC. If BD = 8, DC = 6, and a line through D parallel to AB intersects AC at E, and AE = 5, then this is the length of EC.
What is 3.75? (By proportionality in triangle ABC with DE || AB: DC/BD = EC/AE, so 6/8 = EC/5, thus EC = 3.75)
In right triangle ABC with right angle C, the altitude to hypotenuse AB creates segments of length 4 and 9. This is the length of the altitude.
What is 6? (By geometric mean: altitude = √(4 × 9) = √36 = 6)