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Pythagoras' Theorem: Advanced Smart Worksheet

Smart Worksheet
Grade 11
English

Pythagoras' Theorem: Advanced Smart Worksheet

Activity 1: Prove the Converse of Pythagoras' Theorem

Given triangle $ABC$ with sides

,

, and

. Prove whether triangle $ABC$ is a right triangle using the converse of Pythagoras' Theorem.


Activity 2: Multi-Step Application in 3D Geometry

A rectangular box has dimensions

cm,

cm, and

cm. Calculate the length of the space diagonal (the longest diagonal connecting two opposite corners) using Pythagoras' Theorem.
Show all steps clearly:


Activity 3: Algebraic Manipulation

Given

, solve for

.
Work out each algebraic step:


Activity 4: Proof by Rearrangement

Explain, with a diagram, how rearranging four identical right triangles within a square can be used to visually prove Pythagoras' Theorem. Draw and label your diagram below.



Activity 5: Non-Right Triangles

Given triangle $DEF$ with

,

, and

, determine whether the triangle is right-angled, obtuse, or acute using the law of cosines and the relationship to Pythagoras' Theorem.
Show your calculations:


Activity 6: Missing Side with Irrational Result

A right triangle has legs of length $5$ and

. Find the exact value of the hypotenuse and express your answer in simplest radical form.


Activity 7: Pythagorean Triples Investigation

Find all primitive Pythagorean triples where the hypotenuse is less than

. List at least five such triples and explain why they are considered primitive.


Activity 8: Application in Coordinate Geometry

Given points $A(2, 3)$ and

, use Pythagoras' Theorem to find the distance between $A$ and $B$ in the coordinate plane. Show all steps.


Activity 9: Real-World Problem Solving

A ladder is leaning against a vertical wall. The foot of the ladder is $6$ meters from the wall, and the ladder reaches a point $8$ meters up the wall. What is the length of the ladder? If the top of the ladder slips down by $2$ meters, how far does the foot slide outwards? Show all calculations.


Activity 10: Proof with Similar Triangles

Prove Pythagoras' Theorem using similar triangles. Clearly label the triangles used and provide a step-by-step logical argument.