Question

congruent? ○ yes ○ no; similar? ○ yes ○ no

Explanation:

Step1: Analyze Congruence

Congruent figures have same shape and size. Let's count grid units for sides. Top triangle: vertical side (height) = 3 units, horizontal side (base) = 2 units? Wait, no, looking at the grid, top triangle: from vertex, vertical down 3, horizontal right 2? Wait, no, bottom triangle: horizontal left 3, vertical up 2? Wait, no, let's check coordinates. Wait, maybe better: congruent requires same side lengths and angles. Let's see the two triangles: one is a right triangle with legs (let's say) length 3 and 2? Wait, no, maybe I miscount. Wait, actually, both triangles are right triangles. Let's check the lengths. Top triangle: vertical leg (height) – let's count grid lines. Suppose each grid is 1 unit. Top triangle: from the right angle, vertical up 3, horizontal right 2? No, wait bottom triangle: horizontal left 3, vertical up 2? Wait, no, maybe the legs are 3 and 2, but swapped? Wait, no, congruent triangles must have all corresponding sides equal. Wait, maybe I made a mistake. Wait, actually, let's check the angles. Both are right triangles. Now, check side lengths. Let's assume the grid has each square as 1x1. Top triangle: vertical leg (length) – let's say from the right angle, moving up 3 units, right 2 units? No, bottom triangle: left 3 units, up 2 units? Wait, no, maybe the legs are 3 and 2, but the triangles are congruent? Wait, no, wait the orientation: one is rotated. Wait, congruent triangles can be rotated. Wait, let's count the legs. Top triangle: vertical leg (height) – let's count the number of grid squares. Suppose the top triangle: from the bottom vertex (right angle) up 3, right 2. Bottom triangle: from the right vertex (right angle) left 3, up 2. Wait, no, maybe the legs are 3 and 2, so the hypotenuse would be same. Wait, but maybe I miscounted. Wait, actually, both triangles have legs of length 3 and 2 (assuming grid units), so they are congruent. And since congruent triangles are also similar (similarity ratio 1), so both congruent and similar are yes. Wait, but let's confirm.

Wait, maybe the top triangle: vertical leg is 3, horizontal leg is 2. Bottom triangle: horizontal leg is 3, vertical leg is 2. So they are congruent (since we can rotate one to match the other, and sides are equal: 3, 2, and hypotenuse $\sqrt{3^2 + 2^2} = \sqrt{13}$ for both). So congruent: yes. Similar: yes, because congruent implies similar (ratio 1).

Step2: Analyze Similarity

Similar figures have same shape (same angles) and proportional sides. Since they are congruent, their sides are in ratio 1:1, so they are similar.

Answer:

Congruent? $\boldsymbol{\circ}$ Yes $\quad \circ$ No
Similar? $\boldsymbol{\circ}$ Yes $\quad \circ$ No

(Mark the "Yes" circles for both Congruent and Similar.)