Question

the two triangles are
□ congruent
□ neither
□ similar
because the smaller triangle can be select an answer
and then
select an answer
to get the larger triangle

Explanation:

Step1: Check for congruence

Congruent triangles have equal side - lengths.
For the smaller triangle: vertical side \(y_2 - y_1=5 - 1 = 4\), horizontal side \(x_2 - x_1=5 - 3=2\). Using the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), the hypotenuse \(h_1=\sqrt{4^{2}+2^{2}}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}\).
For the larger triangle: vertical side \(y_2 - y_1=13 - 5 = 8\), horizontal side \(x_2 - x_1=10 - 3 = 7\). The hypotenuse \(h_2=\sqrt{8^{2}+7^{2}}=\sqrt{64+49}=\sqrt{113}\). Since side - lengths are not equal, they are not congruent.

Step2: Check for similarity

For similar triangles, the ratios of corresponding sides are equal.
Let's assume the vertical and horizontal sides are corresponding.
The ratio of vertical sides \(\frac{8}{4}=2\), the ratio of horizontal sides \(\frac{7}{2}=3.5\). But if we consider the right - angled nature and the slope (using the concept of similar right - angled triangles formed by the grid).
The smaller triangle has a vertical leg \(a_1 = 4\) (from \(y = 1\) to \(y = 5\)) and horizontal leg \(b_1=2\) (from \(x = 3\) to \(x = 5\)). The larger triangle has a vertical leg \(a_2=8\) (from \(y = 5\) to \(y = 13\)) and horizontal leg \(b_2 = 6\) (from \(x = 3\) to \(x = 9\)).
The ratio of vertical sides \(\frac{a_2}{a_1}=\frac{8}{4} = 2\), the ratio of horizontal sides \(\frac{b_2}{b_1}=\frac{6}{3}=2\).
The smaller triangle can be dilated (scaled) by a factor of \(2\) (enlarged) to get the larger triangle.

Answer:

The two triangles are similar because the smaller triangle can be dilated (scaled) and then translated (moved without rotation or reflection) to get the larger triangle.