DCA 5 Corrections

Elijah and Benjamin are exploring the magical world of geometry! They have a triangle JKL that they want to enlarge using a special spell centered at the origin. This spell transforms triangle JKL into triangle J’K’L’. Which magical rule best represents the transformation applied to triangle JKL to create triangle J’K’L'?

• (x, y) → (x + 6, y – 3)

• (x, y) → (x + 4.5, y + 3)

Mason and Grace are building a rectangular picture frame for their art project. The length of the frame is 15 inches, and the width is 8 inches. Can you help them find the length of the diagonal of this frame in inches?

• 17 inches

• 15 inches

• 16 inches

• 18 inches

Answer explanation

To find the diagonal of the frame, use the Pythagorean theorem: diagonal = √(length² + width²) = √(15² + 8²) = √(225 + 64) = √289 = 17 inches. Thus, the correct answer is 17 inches.

Kai and Abigail are playing a game in a park where they have a giant rectangle RSTU drawn on the ground. Two shaded regions within the rectangle are squares. Kai measures the length of UP as 4 meters, while Abigail finds the length of UQ to be 8 meters. They need your help to find out the length of PQ in meters. Can you solve it for them?

Emma and Isla are playing a game with shapes. Emma decides to transform rectangle LMNO using the magical rule (x, y) → (-x, y). What kind of magical transformation did Emma perform?

• Translation

• Reflection over the y-axis

• Rotation

• Reflection over the x-axis

Answer explanation

Emma's transformation (x, y) → (-x, y) reflects points across the y-axis, changing the sign of the x-coordinate while keeping the y-coordinate the same. Thus, the correct choice is 'Reflection over the y-axis'.

William is trying to rescue a cat stuck on a ledge! He has a 10-foot ladder and places it against a wall. The bottom of the ladder is 4 feet away from the base of the wall. Can you help William find out which of the following is closest to the distance from the top of the ladder to the base of the wall?

• 6 ft.

• 9 ft.

• 11 ft.

• 14 ft.

Answer explanation

Using the Pythagorean theorem, we have a right triangle with the ladder as the hypotenuse (10 ft), the distance from the wall (4 ft) as one leg, and the height (h) as the other leg. Thus, h = √(10² - 4²) = √(100 - 16) = √84 ≈ 9.2 ft. The closest distance from the top of the ladder to the base of the wall is 6 ft.

Grace and Luna are playing a game on a coordinate grid. They have a magical pentagon that moves around the grid. The pentagon is enchanted to slide 1 unit to the left and 10 units down when they say the magic words. Can you help them figure out which spell (rule) they should use to make this transformation happen?

• (x, y) → (x - 1, y - 10)

• (x, y) → (x + 1, y - 10)

• (x, y) → (x - 1, y + 10)

• (x, y) → (x + 1, y + 10)

Answer explanation

The pentagon moves 1 unit left (decrease x by 1) and 10 units down (decrease y by 10). Therefore, the correct transformation is (x, y) → (x - 1, y - 10).

Nora and Aiden are playing a game with a magical shrinking square! The square is enchanted to shrink by a scale factor of 0.4, with the origin as the center of this magical transformation. If (x, y) represents the location of any point on the original square, can you help them figure out which ordered pair represents the coordinates of the corresponding point on the new, smaller square?

• (20x, 20y)

• (0.4x, 0.4y)

• (x + 20, y + 20)

• (x + 0.4, y + 0.4)

Answer explanation

The square shrinks by a scale factor of 0.4, meaning each coordinate (x, y) is multiplied by 0.4. Thus, the new coordinates are (0.4x, 0.4y), making this the correct choice.

Abigail and Luna are exploring a magical world where objects can change size! They found a mysterious rule book. Can you help them figure out which of the following rules describes the magical dilation spell?

Answer explanation

The magical dilation spell reduces the size of the x-coordinate by a factor of \(\frac{3}{5}\) while keeping the y-coordinate unchanged, as seen in the first choice. The last choice reduces both coordinates, which is also valid.

Liam and Anika are playing a game on a coordinate grid with a rectangle. Which magical transformation will turn their rectangle into something new and exciting, making it NOT congruent to the original rectangle?

• A. A translation 90 units to the right.

• B. A rotation 180° counterclockwise.

• C. A dilation by a scale factor of 3.

• D. A reflection across the x-axis.

Answer explanation

A dilation by a scale factor of 3 enlarges the rectangle, changing its dimensions and making it not congruent to the original. The other transformations (translation, rotation, reflection) preserve congruence.

Olivia and Ethan are designing a triangular garden. They know two sides of the triangle are 7 meters and 24 meters. Can you help them find the length of the third side if it is the longest side of the triangle?

• 25 meters

• 26 meters

• 23 meters

• 24 meters

Answer explanation

To find the longest side of the triangle, we apply the triangle inequality. The sum of the two shorter sides (7m + 24m = 31m) must be greater than the third side. The longest side can be 25m, satisfying this condition.