Question

triangle jkl is graphed on a coordinate plane, as shown. the triangle is then rotated 90° clockwise about the origin, and then translated 10 units to the left and then translated 1 unit up. which of the following statements are true? select all that apply. point k is the same point as k. point l is located in quadrant ii. triangle jkl has an area equal to half the area of △jkl. point j is the same as point l. angles j, j, j, and j are congruent.

Explanation:

Step1: Recall rotation and translation rules

A 90 - degree clock - wise rotation about the origin has the rule $(x,y)\to(y, - x)$. Translation 10 units left means subtract 10 from the x - coordinate and 1 unit up means add 1 to the y - coordinate.

Step2: Analyze each statement

• For the statement "Point K is the same point as K''': Transformations change the position of a point, so this is false.
• For the statement "Point L' is located in Quadrant II": After rotation and translation, we need to find the new coordinates of L. Let's assume $L(x_0,y_0)$. After 90 - degree clock - wise rotation, $L'(y_0,-x_0)$. After further translations, the new point has coordinates $(y_0 - 10,-x_0 + 1)$. Without knowing the exact coordinates of L, we can't be sure it's in Quadrant II, so we can't confirm this.
• For the statement "Triangle J'K'L' has an area equal to half the area of $\triangle JKL$": Rotations and translations are rigid motions. Rigid motions preserve the shape and size of a figure, so the area of $\triangle J'K'L'$ is equal to the area of $\triangle JKL$, and this statement is false.
• For the statement "Point J is the same as point L''': Transformations change the position of points, so this is false.
• For the statement "Angles J, J', J'', and J''' are congruent": Since rotations and translations are rigid motions, they preserve angle measures. So angles J, J', J'', and J''' are congruent, and this statement is true.

Answer:

Angles J, J', J'', and J''' are congruent.