Question

similarity group work
question
when a figure is enlarged, how are corresponding angles related?
how are corresponding lengths related?
materials

• ruler
• calculator
• protractor

explore
photo 1 is an enlargement of photo 2.
1 use a ruler to find the length of \\( \overline { a b } \\) in each photo.
then use a calculator to find the ratio of \\( a b \\) in
photo 1 to \\( a b \\) in photo 2. round to the nearest tenth.
2 use a protractor to find \\( m \angle 1 \\) in each photo.
then find the ratio of \\( m \angle 1 \\) in photo 1 to \\( m \angle 1 \\) in photo 2.
3 continue finding measurements in the photos
and record your results in a table like the one
shown below.

think about it

1. make a conjecture about the relationship between corresponding

lengths when a figure is enlarged.

1. make a conjecture about the relationship between corresponding

angles when a figure is enlarged.

1. suppose an angle in photo 2 has a measure of \\( 35 ^ { \circ } \\). what is the

measure of the corresponding angle in photo 1?

1. extension suppose a segment in photo 1 is 5 centimeters long.

what is the measure of the corresponding segment in photo 2?

Explanation:

When a figure is enlarged (a form of similarity transformation), by the properties of similar figures:

• For corresponding angles: In similar figures, corresponding angles are congruent. This is because the shape of the figure is preserved in an enlargement (only the size changes).
• For corresponding lengths: In similar figures, corresponding lengths are proportional. The ratio of corresponding lengths is called the scale factor.

For question 3: Since corresponding angles in similar figures (Photo 1 is an enlargement of Photo 2, so they are similar) are congruent, if an angle in Photo 2 has a measure of \(35^{\circ}\), the corresponding angle in Photo 1 also has a measure of \(35^{\circ}\).

For question 4: Let the length of the corresponding segment in Photo 2 be \(x\). The ratio of lengths (scale factor) from Photo 1 to Photo 2 is \(k = 1.2\) (from \(AB\) values: \(\frac{AB_{1}}{AB_{2}}=\frac{6}{5}=1.2\)). We know that \(\frac{\text{Length in Photo 1}}{\text{Length in Photo 2}}=k\). Given length in Photo 1 \(l_1 = 5\) cm, then \(\frac{5}{x}=1.2\), so \(x=\frac{5}{1.2}=\frac{50}{12}=\frac{25}{6}\approx4.2\) cm.

Answer:

1. Corresponding lengths are proportional.
2. Corresponding angles are congruent.
3. \(35^{\circ}\)
4. Approximately \(4.2\) cm