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G.8(C)·Geometry·Triangles & TrigonometryReadiness

The Pythagorean Theorem

Use the Pythagorean theorem to find unknown lengths in context.

Students identify right triangles in contextual diagrams, solve for a leg or hypotenuse, and combine a calculated segment with another length to answer the actual question.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Geometry key. Geometry is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutorAll Geometry released practice50 minutes

Learn the concept

Identify the right triangle

The Pythagorean theorem applies to right triangles: a2+b2=c2a^2+b^2=c^2. The legs a and b meet at the right angle. The hypotenuse c is opposite the right angle and is the longest side.

Solve for the correct side

For a missing hypotenuse, use c=a2+b2c=\sqrt{a^2+b^2}. For a missing leg, use a=c2−b2a=\sqrt{c^2-b^2}. Squaring and taking the square root are different steps; the sum of the squares is not the side length.

Return to the context

North-south and east-west travel can form perpendicular legs, with direct separation as the hypotenuse. In a rectangle, a diagonal or interior segment can create a right triangle covering only part of a side. Use segment addition and equal opposite sides to reach the requested length.

Study released STAAR visuals

Use the original graphs, tables, and diagrams from released STAAR questions to practice reading labels, values, and relationships before answering the items below.

Archived released STAAR Geometry EOC · 2013 · Item 17 · 1 point

Rectangle STUW with S top left, T top right, U bottom right, and W bottom left. Point V is on side WU. Segment SV is 20 inches, segment VT is 8 inches, and VU is 5 inches.
Rectangle STUWOriginal visual from the released STAAR test.

Worked example

Rectangle ABCD is labeled in order, with AB along the top. Point E lies on DC. AE = 17 inches, AD = 8 inches, and EC = 4 inches. Find AB.

  1. Triangle ADE is right at D, and AE is its hypotenuse.
  2. Use DE2+82=172DE^2+8^2=17^2, so DE2=289−64=225DE^2=289-64=225.
  3. DE = 15 inches. Add the remaining segment: DC = DE + EC = 15 + 4 = 19 inches.
  4. Opposite sides of a rectangle are congruent, so AB = DC.

19 inches

Watch the lesson

Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

8th Grade Math Applying the Pythagorean Theorem

Five Minute Math · Aaron Daffern, Dallas ISD

Model contextual distances with right triangles and solve for missing lengths.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

  1. 1. A right triangle has hypotenuse 26 centimeters and one leg 10 centimeters. Find the other leg.

    Show answer and reasoning

    24 centimeters

    The missing leg is 262−102=676−100=576=24\sqrt{26^2-10^2}=\sqrt{676-100}=\sqrt{576}=24.

  2. 2. Two delivery robots start together. One travels 14 meters west and the other 23 meters south. Find their direct separation to the nearest tenth.

    Show answer and reasoning

    26.9 meters

    The perpendicular distances are legs, so d=142+232=725≈26.9d=\sqrt{14^2+23^2}=\sqrt{725}\approx26.9. Adding 14 and 23 would give combined travel, not separation.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Geometry EOC · 2013 · Item 17 · 1 point

Rectangle STUW with S top left, T top right, U bottom right, and W bottom left. Point V is on side WU. Segment SV is 20 inches, segment VT is 8 inches, and VU is 5 inches.
Rectangle STUWOriginal visual from the released STAAR test.

Rectangle STUWSTUW is shown. What is STST?

  1. 33 in.
  2. 24 in.
  3. 464\sqrt{464} in.
  4. 361\sqrt{361} in.
Show STAAR answer and explanation

24 in.

In right triangle TUV, TU2=82−52=39TU^2=8^2-5^2=39. Since SW = TU, right triangle SWV gives WV2=202−39=361WV^2=20^2-39=361, so WV = 19. Hence ST=WU=19+5=24ST=WU=19+5=24 inches.

33 in.: Adding 20, 8, and 5 combines slanted segments rather than finding the rectangle length.

464\sqrt{464} in.: This combines squared slanted lengths and omits the two right-triangle relationships.

361\sqrt{361} in.: This equals 19 inches, which is WV; add VU = 5 to get ST.

Ask the tutor about this question

Archived released STAAR Geometry EOC · 2013 · Item 48 · 1 point

Two motorcycles start at the same point. One motorcycle travels 15 km due north and stops. The second motorcycle travels 32 km due west and stops. Which value is closest to the distance between the motorcycles when they stop?

  1. 28.3 km
  2. 47.0 km
  3. 35.3 km
  4. 21.9 km
Show STAAR answer and explanation

35.3 km

The north and west paths are perpendicular legs of a right triangle. Their separation is 152+322=1249≈35.3\sqrt{15^2+32^2}=\sqrt{1249}\approx35.3 km.

28.3 km: This is approximately 322−152\sqrt{32^2-15^2}, which incorrectly treats 32 as the hypotenuse.

47.0 km: This adds the two paths, giving distance through the starting point rather than straight-line separation.

21.9 km: This is approximately 15⋅32\sqrt{15\cdot32}; the Pythagorean theorem adds squared legs.

Ask the tutor about this question
Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

  • Identify the hypotenuse from the right angle rather than diagram orientation.
  • Use the Pythagorean theorem to find a missing leg or hypotenuse.
  • Interpret a calculated length within a larger diagram or perpendicular-travel context.

Instructional sequence

  1. 1

    Engage

    5 min

    Sketch two walkers leaving the same location in perpendicular directions. Compare total travel with the direct separation.

  2. 2

    Direct instruction

    12 min

    Identify the hypotenuse, write the equation, and contrast adding squares to find a hypotenuse with subtracting squares to find a leg.

  3. 3

    Guided practice

    12 min

    Solve missing-side problems in rotated triangles and check that the hypotenuse is longest.

  4. 4

    Context problems

    14 min

    Find a partial rectangle side from a diagonal, then add a remaining segment. Model perpendicular journeys as legs of a right triangle.

  5. 5

    Closure

    7 min

    Complete the exit ticket with a labeled sketch and a final reasonableness check.

Misconceptions to address

““The slanted side is always the hypotenuse.””

Teacher move: The hypotenuse is opposite the right angle. A rotated diagram may make a leg look slanted.

““Always add the two known squares.””

Teacher move: Add when finding the hypotenuse. If the hypotenuse is known, subtract the known leg's square from its square.

““The calculated leg must be the final answer.””

Teacher move: Compare the segment found with the segment requested. A rectangle problem may require adding another piece or using opposite sides.