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AQA GCSE · Question 19 · Geometry and Measures
Here is a right-angled triangle. Use Pythagoras' theorem to show that x = 0.8
Here is a right-angled triangle. Use Pythagoras' theorem to show that x = 0.8
How to approach this question
1. State Pythagoras' theorem: a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle).
2. Identify the sides of the triangle. The hypotenuse c is 1.7 cm. The other two sides, a and b, are x cm and 1.5 cm.
3. Substitute the values into the theorem: x² + 1.5² = 1.7².
4. Calculate the squares: 1.5² = 2.25 and 1.7² = 2.89.
5. The equation becomes x² + 2.25 = 2.89.
6. Rearrange the equation to solve for x²: x² = 2.89 - 2.25.
7. Calculate x² = 0.64.
8. Find x by taking the square root of 0.64: x = √0.64.
9. Calculate the square root: x = 0.8.
Full Answer
a² + b² = c²
x² + 1.5² = 1.7²
x² + 2.25 = 2.89
x² = 2.89 - 2.25
x² = 0.64
x = √0.64
x = 0.8
Pythagoras' theorem states that for a right-angled triangle with sides a, b and hypotenuse c, the following relationship holds:
a² + b² = c²
In the given triangle:
- The hypotenuse (c) is the side opposite the right angle, which is 1.7 cm.
- The other two sides (a and b) are x cm and 1.5 cm.
Substitute these values into the theorem:
x² + 1.5² = 1.7²
Now, calculate the squares:
1.5² = 2.25
1.7² = 2.89
The equation becomes:
x² + 2.25 = 2.89
To find x², subtract 2.25 from both sides:
x² = 2.89 - 2.25
x² = 0.64
Finally, to find x, take the square root of both sides:
x = √0.64
x = 0.8
This shows that x = 0.8.
Common mistakes
✗ Incorrectly identifying the hypotenuse (e.g., setting up as x² + 1.7² = 1.5²).
✗ Adding the squares instead of subtracting when finding a shorter side (e.g., x² = 2.89 + 2.25).
✗ Making a calculation error with the squares or the final square root.
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