Look at the following triangles and answer the questions that following: How many times is: FG longer than BC? JK shorter than BC? EF longer than AB? IJ shorter than AB? Is EFG an enlargement of ABC? Explain your answer. Is IJK a reduction of ABC? Explain your answer.

• 3.1.1
  • Length of FG $= 6$ units
  • Length of BC $= 2$ units
  • Ratio $= \frac{\text{FG}}{\text{BC}} = \frac{6}{2} = 3$
  • Answer: $\boxed{3 \text{ times}}$

• 3.1.2
  • Length of JK $= 1$ unit
  • Length of BC $= 2$ units
  • JK is half the length of BC, meaning it is 2 times shorter.
  • Answer: $\boxed{2 \text{ times}}$ shorter than $\boxed{\text{BC}}$

• 3.1.3
  • Length of EF $= 6$ units
  • Length of AB $= 2$ units
  • EF is 3 times the length of AB.
  • Answer: $\boxed{3 \text{ times}}$ longer than $\boxed{\text{AB}}$

• 3.1.4
  • Length of IJ $= 1$ unit
  • Length of AB $= 2$ units
  • IJ is half the length of AB, meaning it is 2 times shorter.
  • Answer: $\boxed{2 \text{ times}}$ shorter than $\boxed{\text{AB}}$

• 3.1.5
  • Compare corresponding sides: $\frac{\text{FG}}{\text{BC}} = \frac{6}{2} = 3$ and $\frac{\text{EF}}{\text{AB}} = \frac{6}{2} = 3$.
  • Since all corresponding sides are scaled by the same factor (3), $\triangle$EFG is an enlargement of $\triangle$ABC.
  • Answer: $\boxed{\text{Yes}}$
  • Explanation: All corresponding sides of $\triangle$EFG are 3 times longer than the corresponding sides of $\triangle$ABC.

• 3.1.6
  • Compare corresponding sides: $\frac{\text{JK}}{\text{BC}} = \frac{1}{2}$ and $\frac{\text{IJ}}{\text{AB}} = \frac{1}{2}$.
  • Since all corresponding sides are scaled by the same factor ($\frac{1}{2}$), $\triangle$IJK is a reduction of $\triangle$ABC.
  • Answer: $\boxed{\text{Yes}}$
  • Explanation: All corresponding sides of $\triangle$IJK are half the length of the corresponding sides of $\triangle$ABC.

• Let's label the vertices of the trapezium from left to right, top to bottom.
  • The top-left vertex is 2 squares to the left of the mirror line.
  • The bottom-left vertex is 2 squares to the left of the mirror line.
  • The top-right vertex is 1 square to the left of the mirror line.
  • The bottom-right vertex is 4 squares to the left of the mirror line.

• To reflect a point across a vertical mirror line, its horizontal distance from the line remains the same, but its position shifts to the opposite side of the line. The vertical position (height) remains unchanged.
  • The reflected top-left vertex will be 2 squares to the right of the mirror line.
  • The reflected bottom-left vertex will be 2 squares to the right of the mirror line.
  • The reflected top-right vertex will be 1 square to the right of the mirror line.
  • The reflected bottom-right vertex will be 4 squares to the right of the mirror line.

• The reflected shape will be a trapezium, identical in size and shape to the original, but flipped horizontally.
  • The reflected trapezium will have its top-left vertex 1 square right of the mirror line, its top-right vertex 2 squares right of the mirror line, its bottom-left vertex 4 squares right of the mirror line, and its bottom-right vertex 2 squares right of the mirror line.

Answer:

• To reflect the trapezium, plot the new vertices by counting the horizontal distance from each original vertex to the mirror line and then plotting the new vertex the same distance on the opposite side of the mirror line.
  • The vertex that is 2 squares left of the mirror line will be 2 squares right.
  • The vertex that is 1 square left of the mirror line will be 1 square right.
  • The vertex that is 4 squares left of the mirror line will be 4 squares right.
• Connect these new points to form the reflected trapezium.