Question

the triangle will be rotated 180° clockwise around the point (3, 4) to create triangle abc. indicate whether each of the listed features of the image will or will not be the same as the corresponding feature in the original triangle by marking an x in the appropriate box in the table.

	will be the same	will not be the same
the coordinates of c
the perimeter of △abc
the area of △abc
the measure of ∠b
the slope of ac

Explanation:

Step1: Recall Rotation Properties

A 180° rotation around a point \((h,k)\) transforms a point \((x,y)\) to \((2h - x, 2k - y)\). So coordinates of \(A'\), \(C'\) will change (not same as original \(A\), \(C\)).

Step2: Perimeter and Area

Rotation is a rigid transformation, so side lengths (hence perimeter) and area remain same.

Step3: Angle Measure

Rigid transformations preserve angle measures, so \(\angle B'\) measure is same as \(\angle B\).

Step4: Slope of \(A'C'\)

Slope formula: \(m=\frac{y_2 - y_1}{x_2 - x_1}\). After 180° rotation, the slope of \(A'C'\) is same as slope of \(AC\) (since rotation preserves the "steepness" directionally opposite but slope formula gives same value for \((x_1,y_1)\to(2h - x_1, 2k - y_1)\) and \((x_2,y_2)\to(2h - x_2, 2k - y_2)\): \(m'=\frac{(2k - y_2)-(2k - y_1)}{(2h - x_2)-(2h - x_1)}=\frac{y_1 - y_2}{x_1 - x_2}=-\frac{y_2 - y_1}{x_2 - x_1}\)? Wait, no—wait, 180° rotation: vector from center to \(A\) is reversed. Wait, maybe miscalculation. Wait, slope of \(A'C'\): let's take center \((3,4)\). If \(A=(x_A,y_A)\), \(A'=(6 - x_A, 8 - y_A)\); \(C=(x_C,y_C)\), \(C'=(6 - x_C, 8 - y_C)\). Then slope of \(A'C'\) is \(\frac{(8 - y_C)-(8 - y_A)}{(6 - x_C)-(6 - x_A)}=\frac{y_A - y_C}{x_A - x_C}=-\frac{y_C - y_A}{x_C - x_A}\), which is \(-\)slope of \(AC\). Wait, no—wait, slope of \(AC\) is \(\frac{y_C - y_A}{x_C - x_A}\), slope of \(A'C'\) is \(\frac{(8 - y_C)-(8 - y_A)}{(6 - x_C)-(6 - x_A)}=\frac{y_A - y_C}{x_A - x_C}=-\frac{y_C - y_A}{x_C - x_A}\), so slope is same in magnitude and sign? Wait, no: \(\frac{y_A - y_C}{x_A - x_C}=\frac{-(y_C - y_A)}{-(x_C - x_A)}=\frac{y_C - y_A}{x_C - x_A}\). Oh right! Because numerator and denominator both multiplied by \(-1\), so slope is same. So slope of \(A'C'\) is same as slope of \(AC\).

Now, fill the table:

• Coordinates of \(A'\): will not be same
• Coordinates of \(C'\): will not be same
• Perimeter of \(\triangle A'B'C'\): will be same
• Area of \(\triangle A'B'C'\): will be same
• Measure of \(\angle B'\): will be same
• Slope of \(A'C'\): will be same

Answer:

Feature	Will be the same	Will not be the same
Coordinates of \(C'\)		X
Perimeter of \(\triangle A'B'C'\)	X
Area of \(\triangle A'B'C'\)	X
Measure of \(\angle B'\)	X
Slope of \(A'C'\)	X

(Note: In the table, mark 'X' in the appropriate column for each feature as per above analysis.)