Transformations and Congruence: practice solutions, Grade 8 – download the PDF

1 Translate a point ★★★

Add \(-5\) to the \(x\)-coordinate and \(6\) to the \(y\)-coordinate: \(3+(-5)=-2\) and \(-4+6=2\).

Answer: \(P'(-2,\,2)\).

2 Reflect across the axes ★★★

(a) Across the x-axis the sign of \(y\) changes: \((5,\,-2)\).

(b) Across the y-axis the sign of \(x\) changes: \((-5,\,2)\).

3 Half turn about the origin ★★★

A \(180^\circ\) rotation about the origin sends \((x,\,y)\) to \((-x,\,-y)\).

Answer: \(S'(-6,\,1)\).

4 True or false? ★★★

1. False. A translation is rigid, so size does not change.
2. True. Reflections preserve lengths.
3. False. Congruent figures can be in different positions; only size and shape must match.
4. False. Rotations preserve angle measures.

5 Find the translation ★★★

(a) Horizontal change: \(-1-2=-3\). Vertical change: \(1-5=-4\). The vector is \(\langle -3,\,-4\rangle\).

(b) \(B'=(4-3,\ -2-4)=(1,\,-6)\).

6 A reflected rectangle ★★★

A reflection is rigid, so the image is a rectangle 8 m by 5 m.

Perimeter: \(2(8+5)=26\) m. Area: \(8\times5=40\text{ m}^2\). Both are the same as for the original.

7 Angles after a rotation ★★★

The angles of a triangle add to \(180^\circ\): \(\angle C=180-52-67=61^\circ\).

A rotation preserves angles, so \(\angle C'=61^\circ\).

8 Translate a triangle ★★★

Add 6 to each \(x\) and subtract 3 from each \(y\): \(A'(3,\,-2)\), \(B'(5,\,-2)\), \(C'(3,\,1)\).

\(AC=4-1=3\) (same \(x\), so vertical distance) and \(A'C'=1-(-2)=3\). The lengths are equal.

9 A reflected parallelogram ★★★

(a) Change the sign of each \(x\): \(P'(-1,\,1)\), \(Q'(-4,\,1)\), \(R'(-5,\,3)\), \(S'(-2,\,3)\).

(b) Yes. \(PQ\) and \(SR\) are horizontal and both have length 3, so \(PQRS\) is a parallelogram. Reflections preserve parallelism and lengths, so \(P'Q'\parallel S'R'\) with \(P'Q'=S'R'=3\) and the image is a parallelogram too.

10 Rotate a triangle ★★★

Use \((x,\,y)\to(-y,\,x)\): \(A'(-1,\,1)\), \(B'(-1,\,5)\), \(C'(-4,\,1)\).

\(BC=\sqrt{(5-1)^2+(1-4)^2}=\sqrt{25}=5\) and \(B'C'=\sqrt{(-1+4)^2+(5-1)^2}=\sqrt{25}=5\).

11 Match the triangles ★★★

Step 1: rotate \(ABC\) by \(90^\circ\) counterclockwise about the origin: \(A_1(-1,\,1)\), \(B_1(-1,\,5)\), \(C_1(-4,\,1)\).

Step 2: translate by \(\langle 8,\,1\rangle\): \((-1+8,\,1+1)=(7,\,2)=D\), \((-1+8,\,5+1)=(7,\,6)=E\), \((-4+8,\,1+1)=(4,\,2)=F\).

A rotation followed by a translation maps \(ABC\) onto \(DEF\), so \(\triangle ABC\cong\triangle DEF\).

12 Reflect across a vertical line ★★★

\(U\) is \(5-2=3\) units to the right of the line, so \(U'\) is 3 units to the left: \(x=2-3=-1\). The \(y\)-coordinate is unchanged: \(U'(-1,\,3)\).

\(V\) is \(2-(-3)=5\) units to the left, so \(V'\) is 5 units to the right: \(x=2+5=7\). So \(V'(7,\,-2)\).

In general \((x,\,y)\to(4-x,\,y)\).

13 Moves in a video game ★★★

(a) First move: \((3-4,\ 7-2)=(-1,\,5)\). Second move: \((-1+6,\ 5+5)=(5,\,10)\).

(b) \(\langle -4+6,\ -2+5\rangle=\langle 2,\,3\rangle\). Check: \((3+2,\ 7+3)=(5,\,10)\). Two translations in a row make one translation.

14 Turning a wheel ★★★

(a) Spokes are evenly spaced: \(360\div6=60\), so the angle is \(60^\circ\).

(b) \(360\div60=6\) turns of \(60^\circ\).

15 Reflect, then rotate ★★★

(a) After the reflection: \((2,\,-1)\), \((6,\,-1)\), \((2,\,-4)\). After the rotation, \((x,\,y)\to(-y,\,x)\): \(P'(1,\,2)\), \(Q'(1,\,6)\), \(R'(4,\,2)\).

(b) \((2,\,1)\to(1,\,2)\), \((6,\,1)\to(1,\,6)\), \((2,\,4)\to(4,\,2)\): the coordinates are swapped, \((x,\,y)\to(y,\,x)\). This is a reflection across the line \(y=x\).

16 Does the order matter? ★★★

(a) Rotation: \((3,\,2)\to(-2,\,3)\). Reflection: \((-2,\,3)\to(-2,\,-3)\).

(b) Reflection: \((3,\,2)\to(3,\,-2)\). Rotation: \((3,\,-2)\to(2,\,3)\).

The results \((-2,\,-3)\) and \((2,\,3)\) differ, so the order of the transformations matters.

17 A half turn about another center ★★★

(a) \((1,\,0)\to(-1,\,0)\to(3,\,3)\).

(b) The overall rule is \((x,\,y)\to(-x+4,\,-y+3)\). The center is the midpoint of a point and its image: for \(A\) and \(A'(3,\,3)\) it is \(\left(\dfrac{1+3}{2},\,\dfrac{0+3}{2}\right)=(2,\,1.5)\). Rule: \((x,\,y)\to(4-x,\ 3-y)\), a \(180^\circ\) rotation about \((2,\,1.5)\).

18 Drone delivery ★★★

(a) The distance is \(\sqrt{6^2+8^2}=\sqrt{100}=10\) units.

(b) \(10\times5=50\) meters (about 164 feet).

(c) Yes. In a translation every point moves the same distance in the same direction.

19 Which triangle is congruent? ★★★

(a) Rigid motions preserve lengths and angles: \(DE=7\) in., \(EF=9\) in., \(\angle E=40^\circ\) (about 17.8 cm and 22.9 cm).

(b) No. If \(XYZ\) were congruent to \(ABC\) with \(Y\) matching \(B\), then \(\angle Y\) would be \(40^\circ\). Since \(45^\circ\ne40^\circ\), no rigid motion maps one onto the other.

20 Not a rigid motion ★★★

(a) \(O'(0,\,0)\) and \(N'(6,\,8)\).

(b) \(ON=\sqrt{9+16}=5\) and \(O'N'=\sqrt{36+64}=10\). The distance doubles, so lengths are not preserved: the rule is not rigid.

(c) Yes. It adds 3 to every \(y\), which is the translation \(\langle 0,\,3\rangle\).