How to show a value in a rectangle using surds

Expert intro: In geometry, ‘show that’ usually means substitute the given expressions and simplify until the target value appears. The key is keeping the algebra tidy.

Detailed walkthrough

To show that $x=2\sqrt{7}$, you usually use the information given in the rectangle and substitute it into the relevant expression.

A common route is:

1. Write down the expression for the side, diagonal, or unknown given in the diagram.
2. Substitute the known surd lengths.
3. Simplify using standard surd rules.
4. Check that the result matches $2\sqrt{7}$.

If the rectangle involves a relation such as the Pythagorean theorem, then the proof often looks like

$\text{(long side)}^2 = \text{(short side)}^2 + \text{(diagonal)}^2,$

followed by substitution and simplification until $x$ isolates to

$x=2\sqrt{7}.$

What the marker wants to see

• clear substitution
• correct squaring or expansion
• a clean final step showing $x=2\sqrt{7}$

If you want, I can also write this out in full exam style once the rectangle diagram or the given side lengths are provided.

💡 Pitfall guide

A frequent problem is skipping the substitution step and jumping straight to the answer. In proof questions, that can lose marks even if the final value is right. Another trap is forgetting that when you square a surd, $\left(\sqrt{7}\right)^2=7$, not $\sqrt{49}$ written awkwardly.

🔄 Real-world variant

If the unknown were not a length but a diagonal or an area-related expression, the same strategy would still apply, but the equation would change. For example, a diagonal problem might need $d^2=a^2+b^2$, while an area problem would use $A=lw$. The target value may still be $2\sqrt{7}$, but the route depends on the diagram.

🔍 Related terms

Pythagorean theorem, rectangle proof, surd algebra