Flashcards · J560 · Topic 9

Congruence and similarity flashcards

21 cards on Topic 9, subtopics 9.01 to 9.04. Pick a subtopic or study them all.

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What must you give to describe a transformation fully?: Reflection: the mirror line. Rotation: centre, angle and direction. Translation: a column vector. Reflection, rotation and translation keep lengths and angles, so the image is congruent.
Reflection in y = x and in y = −x: y = x: swap the coordinates, (x, y) → (y, x). y = −x: swap and change both signs, (x, y) → (−y, −x).
Rotation about the origin through 90° clockwise and 90° anticlockwise: 90° clockwise: (x, y) → (y, −x). 90° anticlockwise: (x, y) → (−y, x). 180°: (x, y) → (−x, −y).
A translation maps (7, 2) to (3, 6). Find the column vector.: vector = image − object: top 3 − 7 = −4, bottom 6 − 2 = 4, so (−4 over 4). The inverse translation is (4 over −4).
Invariant points of a reflection, rotation and translationHigher: Reflection: every point on the mirror line. Rotation: the centre. Translation: none.
What single transformation is a reflection in two parallel lines?Higher: A translation, at right angles to the lines, of twice the gap between them, from the first line towards the second.

Congruent shapes: Exactly the same shape and size: one can be moved onto the other by translation, rotation or reflection. (Similar shapes have the same shape but may differ in size.)
The four conditions for congruent triangles: SSS: three pairs of sides. SAS: two sides and the included angle. ASA: two angles and a corresponding side. RHS: right angle, hypotenuse and one other side.
Which conditions do not prove congruence?: AAA (same shape, size may differ) and SSA, two sides with a non-included angle (can give two different triangles).
How do you set out a congruence proof?: 1. Name the triangles with matching letters. 2. State three equal pairs, each with a reason. 3. Name the condition and say the triangles are congruent. 4. State what follows.
An isosceles triangle has apex angle 56°. Find each base angle.: Base angles are equal: (180° − 56°) ÷ 2 = 62°.

Column vector from A to B: Top: x of B − x of A. Bottom: y of B − y of A. The top number is right (+) or left (−); the bottom is up (+) or down (−).
Vector rules: BA, AB + BC, and AB in terms of OA and OB: BA = −AB (same size, opposite direction). AB + BC = AC. AB = OB − OA.
When are two vectors parallel?: When one is a number times the other: a = k × b. A negative k means opposite directions.
How do you prove points are collinear with vectors?Higher: Show that two vectors from a shared point are multiples of each other, and state the shared point.
P divides AB in the ratio m : n. Write AP in terms of AB.Higher: AP = m/(m + n) × AB

Tests for similar triangles: Any one: all three pairs of sides in the same ratio; two pairs of sides in the same ratio with equal included angles; or two pairs of equal angles (the third must then be equal too).
How do you find a missing length in similar shapes?: Find the scale factor from a pair of matching sides (new ÷ original). Multiply to go from smaller to larger, divide to go from larger to smaller. Check the size makes sense.
Describing an enlargement fully, and finding its centre: Give the word enlargement, the scale factor and the centre. To find the centre, draw lines through pairs of matching vertices: they meet at the centre.
Enlarge (8, 6) with centre (2, 2) and scale factor −1/2Higher: Steps from the centre: right 6, up 4. Multiply by −1/2: left 3, down 2. Image: (−1, 0). A negative scale factor puts the image on the other side of the centre, turned 180°.
Lengths in the ratio 3 : 5. Ratio of areas and of volumes?Higher: Areas: 3² : 5² = 9 : 25. Volumes: 3³ : 5³ = 27 : 125. General: length factor k gives area factor k² and volume factor k³.