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Pythagoras’ Theorem

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Foundational skills
Identify the hypotenuse
\[ \text{Hypotenuse: } p, q \text{ or } r? \]
Find the side opposite the right angle.
Square a decimal
\[ 6.5^2 = 42.25 \]
Square a decimal without doubling it.
Square root to 1 d.p.
\[ \sqrt{58} = 7.6 \text{ (1 d.p.)} \]
Find a square root and round it to 1 decimal place.
Add or subtract?
\[ \text{Add or subtract?} \]
Decide whether to add or subtract the squares.
Finding the hypotenuse
Hypotenuse: whole-number answer
\[ x^2 = 6^2 + 8^2 \]
Add the squares of the shorter sides, then square root.
Hypotenuse: answer to 1 d.p.
\[ x = \sqrt{5^2 + 9^2} \]
Find the hypotenuse and round to 1 decimal place.
Hypotenuse: decimal sides
\[ x^2 = 3.2^2 + 4.5^2 \]
Find the hypotenuse when the sides are decimals.
Finding a shorter side
Shorter side: whole-number answer
\[ x^2 = 13^2 – 5^2 \]
Subtract the squares, then square root.
Shorter side: answer to 1 d.p.
\[ x = \sqrt{11^2 – 6^2} \]
Find a shorter side and round to 1 decimal place.
Shorter side: decimal sides
\[ x^2 = 8.4^2 – 2.7^2 \]
Find a shorter side when the sides are decimals.
Is it right-angled?
Whole-number sides
\[ 7, 24, 26: \text{ right-angled?} \]
Use Pythagoras to test whether a triangle is right-angled.
Decimal sides
\[ 1.5, 2, 2.5: \text{ right-angled?} \]
Test a triangle with decimal sides.
Pythagoras in shapes
Diagonal of a rectangle
\[ \text{Diagonal of } 6 \times 9 \]
The diagonal is the hypotenuse of a right-angled triangle.
Height of an isosceles triangle
\[ \text{Height: } 10, 10, 12 \]
Halve the base, then use Pythagoras.
Slant side of a trapezium
\[ \text{Slant side of trapezium} \]
Use the difference between the parallel sides.
Side of a square from its diagonal
\[ x^2 + x^2 = 10^2 \]
Both shorter sides are equal, so solve x² + x² = d².
Coordinates
Length of a line on a grid
\[ \text{Length of } AB \text{ (grid)} \]
Count across and up, then use Pythagoras.
Distance between two points
\[ (-2, 3) \text{ to } (4, -5) \]
Find the differences in x and y, then use Pythagoras.
Extensions
Two triangles sharing a side
\[ \text{Find } BD \text{ first} \]
Find the shared side first, then use it in the second triangle.
Perimeter of a right-angled triangle
\[ \text{Perimeter: legs } 5, 12 \]
Find the missing side, then add all three sides.
Area of an isosceles triangle
\[ \text{Area: } 13, 13, 10 \]
Find the height with Pythagoras, then the area.
Ladder against a wall
\[ \text{Ladder } 5\text{m, foot } 1.5\text{m} \]
The ladder is the hypotenuse.
Compass journeys
\[ 5\text{km N, then } 12\text{km E} \]
Find how far from the start, in a straight line.
Timer (Optional)
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