Study guide: 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.
Ideas to explain in your own words
- A dilation centered at the origin multiplies both coordinates by the same scale factor. The image stays on the original origin ray while its distance changes multiplicatively. The center matters: multiplying coordinates directly assumes center (0,0).
- The hypotenuse square equals the sum of leg squares. If one leg is unknown, subtract the known leg’s square from the hypotenuse square before taking the positive square root. Identify the longest side as the hypotenuse.
- 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. This is the theorem’s converse. Use the largest side as the candidate hypotenuse.
- A box’s base diagonal and height form a second right triangle. Applying the theorem twice gives a space-diagonal square equal to the sum of all three squared perpendicular dimensions. The route along edges differs from the straight interior diagonal.
Useful vocabulary
- dilation
- Scaling distances from a specified center by a common factor.
- hypotenuse
- The side opposite a right angle.
- converse
- A statement formed by reversing hypothesis and conclusion.
Watch your reasoning
- The center matters: multiplying coordinates directly assumes center (0,0). A dilation with factor three adds no fixed coordinate increment; it also scales lengths without preserving congruence unless the factor is one.
- Identify the longest side as the hypotenuse. Adding the squares of a hypotenuse and a leg does not recover the other leg; the final side length uses a positive root and consistent units.
- Use the largest side as the candidate hypotenuse. Triangle inequality alone establishes a possible triangle rather than a right triangle; a near-equality from rounded measurements needs separate error analysis.
A worked review example
Complete supplied guide examples: Scale Coordinates from the Origin — Dilate (2,3) by scale factor 3 about the origin. Candidate answers: (A) (0,1) (B) (2,9) (C) (5,6) (D) (6,9). Multiply both coordinates by 3 because the dilation center is the origin. The point moves along its origin ray to triple its original distance. Subtract Squares to Recover a Leg — A right triangle has hypotenuse 10 cm and one leg 8 cm. Find its other leg. Candidate answers: (A) 164 cm (B) 18 cm (C) 2 cm (D) 6 cm. For the missing leg z, z²+(8)²=(10)². Thus z²=36; take the positive root, z=6 cm. Test a Right Angle from Three Sides — A valid triangle has complete side lengths (6, 8, 10). 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. The largest side is 10. Compare 6²+8²=100 with 10²=100. They agree, so the Pythagorean converse establishes that the triangle is right. Combine Two Perpendicular Right-Triangle Models — A rectangular box has dimensions 4,6,12 cm. Find its interior corner-to-opposite-corner diagonal. Candidate answers: (A) 13 cm (B) 14 cm (C) 15 cm (D) 28 cm. First square the base diagonal: (4)²+(6)²=52. Then add height²=144, so the space diagonal²=196 and its positive root is 14.
Make a practice plan
- Study the four methods in the listed order; attempt each independent check before opening its explanation.
- Complete each lesson’s three changed-input questions and record the operation or reasoning step behind any correction.
- Take the four-question unit review, then solve the transfer task with written justifications.
- After correcting errors, attempt the separate unit exam and revisit the linked lesson for any unresolved method.