Practice Maths

Applications of Pythagoras’ Theorem

Key Ideas

Key Terms

Converse of Pythagoras’ theorem
If a2 + b2 = c2, then the triangle is right-angled (with c the longest side).
Distance on a coordinate plane
D = √[(x2−x1)2 + (y2−y1)2]
3D box diagonal
D = √(l2 + w2 + h2). Apply Pythagoras twice.
Real-world applications
Include: ladders, distances across fields, heights of poles/trees, navigation, and construction.
ApplicationFormula
Coordinate distanced = √[(Δx)2 + (Δy)2]
3D space diagonald = √(l2 + w2 + h2)
Converse testRight-angled if a2 + b2 = c2 (c = longest)
Hot Tip For the converse test, always compare a2+b2 with c2 where c is the longest side. If equal: right-angled. If a2+b2 > c2: acute. If a2+b2 < c2: obtuse.

Worked Example — 3D Diagonal of a Box

Question: Find the diagonal of a box with length 6 m, width 2 m, and height 3 m.

Step 1 — Base diagonal. dbase = √(62 + 22) = √(36+4) = √40

Step 2 — Space diagonal. d = √(dbase2 + h2) = √(40 + 9) = √49 = 7 m

The Importance of Drawing a Diagram

Application problems rarely present a right-angled triangle directly — you must recognise one hidden within a practical context and draw it. For any Pythagoras application, draw a labelled diagram first, mark the right angle, label the known sides with their values, and label the unknown side (usually as x or using its geometric name). A clear diagram prevents the most common error in application problems: using the wrong sides in the formula.

Diagonal of a Rectangle

A rectangle's diagonal divides it into two right-angled triangles. The two shorter sides of the triangle are the width and height of the rectangle; the diagonal is the hypotenuse. For a rectangle 9 m wide and 12 m tall: diagonal2 = 92 + 122 = 81 + 144 = 225, so diagonal = √225 = 15 m. Real-world use: electricians and builders use this to find the length of cable needed to run diagonally across a wall or ceiling.

Height of an Isosceles Triangle

To find the height of an isosceles triangle, draw the perpendicular from the apex to the base. This creates two congruent right-angled triangles. The hypotenuse of each is the slant side of the original triangle; one leg is half the base; the other leg is the height you need to find. For an isosceles triangle with equal sides of 10 cm and base 12 cm: each right triangle has hypotenuse 10 and one leg 6. Height2 = 102 − 62 = 100 − 36 = 64, so height = 8 cm.

Distance Between Two Coordinate Points

When two points are plotted on a coordinate plane, the straight-line distance between them is the hypotenuse of a right triangle. The horizontal distance (difference in x-coordinates) and vertical distance (difference in y-coordinates) are the two shorter sides. This is exactly how the distance formula √((x2−x1)2 + (y2−y1)2) is derived. Drawing the triangle on the grid makes the application of Pythagoras obvious.

Key tip: For 3D problems (like finding the space diagonal of a box), apply Pythagoras twice in two stages. First, find the diagonal of the rectangular base using the length and width. Then use that diagonal as one leg and the height as the other leg in a second right triangle to find the space diagonal. Label your working clearly so you know which triangle you are solving at each stage.

3D Applications: The Box Diagonal

The space diagonal of a rectangular box with dimensions l × w × h is found in two steps. Step 1: find the base diagonal dbase = √(l2 + w2). Step 2: the space diagonal d = √(dbase2 + h2) = √(l2 + w2 + h2). For a box that is 3 m × 4 m × 12 m: dbase = √(9 + 16) = 5, then d = √(25 + 144) = √169 = 13 m. This formula is used in engineering and architecture to check that components will fit diagonally inside a room or container.

Mastery Practice

  1. Find the exact distance between each pair of points. Show horizontal and vertical differences clearly. Fluency

     Point 1Point 2Distance (exact)
    (a)(0, 0)(5, 12) 
    (b)(2, 3)(6, 6) 
    (c)(−3, 1)(1, 4) 
    (d)(4, −2)(4, 7) 
    (e)(−1, −2)(3, 1) 
    (f)(0, −4)(3, 0) 
    (g)(1, 1)(7, 9) 
    (h)(−2, 3)(4, −5) 

  2. Use the converse of Pythagoras’ theorem to determine whether each triangle is right-angled. Write Yes or No and show your working. Fluency

     Three sidesa2+b2 vs c2Right-angled?
    (a)6, 8, 10  
    (b)5, 7, 9  
    (c)9, 40, 41  
    (d)8, 10, 13  
    (e)20, 21, 29  
    (f)11, 60, 61  

  3. Complete the Pythagorean triple table. Find each missing value and verify with a2+b2=c2. Fluency

     abc (hypotenuse)
    (a)34 
    (b)5 13
    (c) 2425
    (d)815 
    (e)9 41
    (f) 2129
    (g)1160 
    (h)12 37

  4. True or False? Justify each answer. Fluency

    1. If a2+b2 = c2, the triangle must have a right angle at the vertex between sides a and b.
    2. The distance formula is just Pythagoras’ theorem applied to horizontal and vertical differences.
    3. A triangle with sides 6, 8, and 11 is right-angled.
    4. The space diagonal of a cube with side length s is s√3.
    5. For any two points on a coordinate grid, the distance is always a whole number.

  5. Navigation and walking problems. Understanding

    Sailing trip. A sailing boat travels 12 km north and then 5 km west. A second boat travels 400 m due north and then 300 m due east.
    1. Find the exact straight-line distance from the start to the finish of the first boat’s trip.
    2. Find the straight-line distance for the second boat. Is it a Pythagorean triple?
    3. A third boat travels d km east then 15 km north and ends up 17 km from its start. Find d.

  6. Diagonal of a square paddock. Understanding

    Square paddock. A square paddock has a diagonal of 200 m. A communications tower casts a shadow 30 m long on flat ground, and a cable from the top of the tower to the tip of the shadow is 50 m long.
    1. Find the side length of the paddock in simplest exact surd form.
    2. Find the area of the paddock.
    3. Find the height of the communications tower.

  7. Reasoning with Pythagoras’ theorem. Understanding

    1. A triangle has sides 2√3, 4, and 2√7. Is it right-angled? Show all working with exact values.
    2. Two points are at A(1, 2) and B(7, k). If the distance AB = 10, find the two possible values of k.
    3. Explain algebraically why the diagonal of a square with side length s is always s√2.

  8. 3D space diagonals. Understanding

    Rectangular boxes. Use Pythagoras’ theorem twice to find the space diagonal: first find the base diagonal, then combine with the height.
     LengthWidthHeightSpace diagonal (exact)
    (a)3 m4 m12 m 
    (b)2 m3 m6 m 
    (c)4 cm4 cm4 cm (cube) 
    (d)6 cm6 cm7 cm 

  9. Pole cables and fitting a rod. Problem Solving

    Support cables. A pole 8 m tall is secured with two cables. Cable A runs from the top to a point 6 m from the base. Cable B runs from the top to a point 15 m from the base on the opposite side.
    1. Find the length of Cable A.
    2. Find the length of Cable B.
    3. Cable costs $4.50 per metre. Find the total cost of both cables.
    4. A moving company needs to fit a 4.5 m pole diagonally in a van 2.4 m long, 1.6 m wide, and 1.8 m high. Will the pole fit? Show full working.

  10. Coordinate geometry: triangle and altitude. Problem Solving

    Triangle ABC. Triangle ABC has vertices A(0, 0), B(4, 0), and C(0, 3).
    1. Find the lengths AB, AC, and BC. Show all working.
    2. Use the converse of Pythagoras to show triangle ABC is right-angled. State where the right angle is.
    3. Find the area of triangle ABC using the legs as base and height.
    4. Find the altitude from A to hypotenuse BC by using the area formula a second way: Area = ½ × BC × altitude.
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