The Life Starter
Sign up
Grade 8 · unit review

Review: Dilations and Right-Triangle Structure

Dilations change lengths by a common factor and preserve shape. Right-triangle relationships connect squared perpendicular lengths; their converse tests whether supplied side lengths support a right angle, including in a three-dimensional diagonal model.

Download reviews and answer keys

Bring the unit skills together

Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. Dilate (7,8) by scale factor 3 about the origin.
Need a hint?
  • A dilation centered at the origin multiplies both coordinates by the same scale factor.
  • The center matters: multiplying coordinates directly assumes center (0,0).
2. A right triangle has hypotenuse 35 cm and one leg 28 cm. Find its other leg.
Need a hint?
  • The hypotenuse square equals the sum of leg squares.
  • Identify the longest side as the hypotenuse.
3. A valid triangle has complete side lengths (21, 28, 36). Which justified conclusion is correct?
Need a hint?
  • For a valid triangle, equality between the largest side’s square and the sum of the other two squares establishes a right angle opposite the largest side.
  • Use the largest side as the candidate hypotenuse.
4. A rectangular box has dimensions 14,21,42 cm. Find its interior corner-to-opposite-corner diagonal.
Need a hint?
  • A box’s base diagonal and height form a second right triangle.
  • The route along edges differs from the straight interior diagonal.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. Dilate (9,10) by scale factor 3 about the origin. Candidate answers: (A) (12,13) (B) (27,30) (C) (3,3) (D) (9,30). 2. A right triangle has hypotenuse 45 cm and one leg 36 cm. Find its other leg. Candidate answers: (A) 27 cm (B) 3321 cm (C) 81 cm (D) 9 cm. 3. A valid triangle has complete side lengths (27, 36, 46). Which justified conclusion is correct? Candidate answers: (A) Cannot decide from complete side lengths (B) Not right: squared-side equality fails (C) Right triangle, because triangle inequality holds (D) Right triangle: squared-side equality holds. 4. A rectangular box has dimensions 18,27,54 cm. Find its interior corner-to-opposite-corner diagonal. Candidate answers: (A) 126 cm (B) 62 cm (C) 63 cm (D) 64 cm.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. (27,30). Multiply both coordinates by 3 because the dilation center is the origin. The point moves along its origin ray to triple its original distance. 2. 27 cm. For the missing leg z, z²+(36)²=(45)². Thus z²=729; take the positive root, z=27 cm. 3. Not right: squared-side equality fails. The largest side is 46. Compare 27²+36²=2025 with 46²=2116. They differ, so the Pythagorean converse establishes that the triangle is not right. 4. 63 cm. First square the base diagonal: (18)²+(27)²=1053. Then add height²=2916, so the space diagonal²=3969 and its positive root is 63.

A different response can also meet the criteria. Compare the evidence and reasoning, rather than copying the wording.