Question

practice 8.2.06
problem 1
which graph shows a pair of similar figures?

Explanation:

Step1: Recall the definition of similar figures

Similar figures have corresponding angles equal and corresponding sides in proportion.

Step2: Analyze the first set of parallelograms

For the first set of parallelograms (top - left), the side - length ratios are not consistent. Let's assume the base and height of the first parallelogram. If we consider the horizontal and vertical displacements of the vertices. For the blue parallelogram, if we assume the base (horizontal side) length is \(a\) and height (vertical side) length is \(b\). For the green parallelogram, the base length is \(c\) and height is \(d\). The ratio \(\frac{a}{c}
eq\frac{b}{d}\) (by counting the grid - squares, if the blue parallelogram has base \(2\) units and height \(1\) unit, and the green parallelogram has base \(2\) units and height \(3\) units, \(\frac{1}{3}
eq\frac{2}{2}\)).

Step3: Analyze the second set of rectangles

For the second set of rectangles (middle). Let the side lengths of the smaller rectangle (green) be \(x\) and \(y\), and the side lengths of the larger rectangle (blue). By counting the grid - squares, if the green rectangle has length \(2\) units and width \(2\) units, and the blue rectangle has length \(3\) units and width \(4\) units. The ratio of lengths \(\frac{2}{3}\) and the ratio of widths \(\frac{2}{4}=\frac{1}{2}\), \(\frac{2}{3}
eq\frac{1}{2}\).

Step4: Analyze the third set of triangles

For the third set of triangles (bottom). Let the side lengths of the larger triangle (green) be \(A\), \(B\), \(C\) and the side lengths of the smaller triangle (blue). By counting the grid - squares (using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), but since they are right - angled triangles on a grid, if the larger right - angled triangle has legs \(4\) units and \(4\) units (by counting grid - squares), and the smaller right - angled triangle has legs \(2\) units and \(2\) units. The ratio of corresponding sides is \(\frac{4}{2}=\frac{4}{2} = 2\). Also, all right - angles are equal (\(90^{\circ}\)), and for non - right angles, since the triangles are right - angled and the sides are in proportion, the other angles (using \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), for the larger triangle \(\tan\theta = 1\) (\(\theta = 45^{\circ}\)) and for the smaller triangle \(\tan\theta=1\) (\(\theta = 45^{\circ}\)).

Answer:

The third graph (the set of triangles) shows a pair of similar figures.