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Similar Right Triangles Worksheets

Use an altitude to the hypotenuse to find missing lengths in a right triangle, with exact radical answers or rational-only practice.

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Use the altitude and hypotenuse segments. Keep exact answers in simplest form.

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Name the whole hypotenuse and both parts

The altitude from the right angle creates three similar right triangles. Use h² = pq, a² = cp and b² = cq with c = p + q.

Learning objective

What this activity builds

Apply the geometric-mean identities that follow from three similar right triangles.

Worked example

An altitude divides a right triangle hypotenuse into lengths 9 and 16. Find the altitude.

12.

The altitude h satisfies h² = 9 × 16 = 144, so the positive length h is 12.

Teaching notes

Make the practice count

  • Identify the altitude foot before assigning the segments.
  • The whole hypotenuse is the sum of its two segments.
  • A leg squared equals the hypotenuse times its adjacent hypotenuse segment.

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Math SheetNameScoreTeacherDateSimilar Right TrianglesUse the altitude-to-hypotenuse relationships. Leave exact answers simplified; sqrt(n) means the square root of n.1.Find h.p = 16q = 24h = xACBDh =2.Find q.p = 4q = xh = 2sqrt(6)ACBDq =Diagrams are not to scale.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDateSimilar Right TrianglesUse the altitude-to-hypotenuse relationships. Leave exact answers simplified; sqrt(n) means the square root of n.3.Find a.p = 10c = 25a = xACBDa =4.Find the total hypotenuse c.p = 6c = xa = 3sqrt(10)ACBDc =Diagrams are not to scale.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDateSimilar Right TrianglesUse the altitude-to-hypotenuse relationships. Leave exact answers simplified; sqrt(n) means the square root of n.5.Find h.p = 20q = 30h = xACBDh =6.Find q.p = 12q = xh = 6sqrt(6)ACBDq =Diagrams are not to scale.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDateSimilar Right TrianglesUse the altitude-to-hypotenuse relationships. Leave exact answers simplified; sqrt(n) means the square root of n.7.Find a.p = 2c = 5a = xACBDa =8.Find the total hypotenuse c.p = 14c = xa = 7sqrt(10)ACBDc =Diagrams are not to scale.math-sheet.comPage 4 of 4

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