Combine Two Perpendicular Right-Triangle Models
A box’s base diagonal and height form a second right triangle.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: 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.
- Independent check inputs: A rectangular box has dimensions 6,9,18 cm. Find its interior corner-to-opposite-corner diagonal. Candidate answers: (A) 20 cm (B) 21 cm (C) 22 cm (D) 42 cm.
Learn the idea
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. 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.
Here is the method on the supplied teaching case. 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.
The route along edges differs from the straight interior diagonal. Use perpendicular box dimensions and add their squares; adding the three lengths instead gives a longer edge-following path.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: 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. Solution: 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. The final answer is 14 cm. The route along edges differs from the straight interior diagonal. Use perpendicular box dimensions and add their squares; adding the three lengths instead gives a longer edge-following path.
Check your reasoning
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Helpful tips
- A box’s base diagonal and height form a second right triangle.
- The route along edges differs from the straight interior diagonal.
See the answer and explanation
21 cm
First square the base diagonal: (6)²+(9)²=117. Then add height²=324, so the space diagonal²=441 and its positive root is 21.
Take it further
Changed-input task: 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. Show the mathematical reason, not only the selected answer. Expected result: 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.
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