Massachusetts Grade 8 Math

Why is the missing leg five units?

A right triangle has one leg of length 12 units and a hypotenuse of length 13 units. Find the other leg. Explain why subtracting 12 from 13 does not solve this problem, and check the answer using areas of squares on the sides. The right angle is given, and the 13-unit side is opposite it.

Explain the solution

For a right triangle, the squares of the leg lengths sum to the square of the hypotenuse. If the missing leg is x, then x squared plus 12 squared equals 13 squared. Since 12 squared is 144 and 13 squared is 169, x squared is 25. A side length is positive, so x is 5 units. Check: 25 plus 144 equals 169. Subtracting the original lengths gives one unit, but the theorem compares squared lengths rather than a straight-line sum of two sides. Three squares built on the triangle’s sides have areas 25, 144, and 169 square units, linking the equation to an area relationship.

Notice the reasoning

The hypotenuse is the side opposite the right angle, not merely the side that looks horizontal in a sketch. The equation must use that side as the squared total. This calculation applies because the triangle is right; it should not be used indiscriminately for every triangle. The converse asks a different question: if three positive side lengths form a triangle and the two smaller squared lengths sum to the largest squared length, then the triangle is right. Checking the 5–12–13 lengths illustrates that condition, but one numerical example is not a general justification of the theorem or its converse.

Try a related task

Draw squares on the sides of a different right triangle and use a teacher-selected rearrangement or area argument to explain why the relationship holds generally. Then compare proposed side lengths 5, 12, and 14: 25 plus 144 is not 196, so these do not describe a right triangle with 14 as the longest side. Ask what information would justify using the theorem in a real measurement problem. The Massachusetts requirement includes analyzing and justifying both the theorem and its converse. Keep that reasoning alongside missing-side calculations and choose related geometry practice according to which part needs further work.

Selected references from Massachusetts mathematics standards

The Massachusetts mathematics standards directory links the current mathematics document, adopted in March 2017 and currently listed in the state’s frameworks directory. These selected references identify the worked activity’s focus. The expectations include numbered requirements, examples, and mathematical practices. Read the full document and confirm the school’s current instructional sequence.

Sources checked October 6, 2026. Read complete requirements, clarifications, limitations, and mathematical practices alongside the school’s sequence. This selected activity is an entry point for instruction.

Connect the explanation with practice

Use the lesson, worksheet, and quiz directory to choose a related skill. The existing resource links remain available; open each activity’s directions to understand its timer, tools, question count, and scoring.

Grade 8 is within the statewide MCAS mathematics assessment range. MCAS is untimed within a regular school day. Independently published practice may use its own timer, tools, question count, and scoring; these are not official administration rules or a prediction of a state performance level. Read Massachusetts’s assessment overview.

Keep one example of the student’s reasoning with the practice result. Choose a smaller model, a related question, or a new context according to the difficulty revealed in the worked activity.