Question

select all the correct locations on the coordinate plane.
select all the triangles that are similar to \\( \triangle a b c \\).

Explanation:

Step1: Calculate side - length ratios

For \(\triangle ABC\), assume \(AB = 2\) units (vertical), \(BC = 2\) units (horizontal). The ratio of vertical to horizontal side - length is \(\frac{AB}{BC}=1\).

Step2: Check other triangles

• For \(\triangle DEF\): \(DE = 2\) units (vertical), \(EF = 2\) units (horizontal). \(\frac{DE}{EF}=1\).
• For \(\triangle XYZ\): \(XY = 4\) units (vertical), \(YZ\) has a horizontal component such that the ratio of vertical to horizontal (by counting grid - units) is \(1\).
• For \(\triangle JKL\): \(JK = 1\) unit (horizontal), \(KL = 1\) unit (vertical). \(\frac{KL}{JK}=1\).
• For \(\triangle MNO\): \(MN = 9\) units (vertical), \(NO = 6\) units (horizontal). \(\frac{MN}{NO}=\frac{3}{2}

eq1\).

• For \(\triangle PQR\): \(PQ = 12\) units (vertical), \(QR = 10\) units (horizontal). \(\frac{PQ}{QR}=\frac{6}{5}

eq1\).

Since similar triangles have proportional corresponding sides and for right - angled triangles (all these are right - angled as they are formed by grid lines), if the ratio of the vertical side to the horizontal side is the same, the triangles are similar (by the Side - Angle - Side similarity criterion as the right angle is common).

Answer:

\(\triangle DEF\), \(\triangle XYZ\), \(\triangle JKL\)