Question

In the given figure, $D EFG$ is a square. $△ A BC$ is right angle triangle with $∠ A = 90°$ . Prove that $A G \times D G = A F \times D B$ .

Approach

In a right triangle $A BC$ with $∠ A = 9 0^{\circ}$ , $D EFG$ is a square inscribed such that $D E ∥ BC$ . Since $GF ∥ BE$ , we have $∠ GB D = ∠ A GF$ (corresponding angles). Thus $△ G A F \sim △ B D G$ (AA similarity). From similarity, $\frac{A G}{D B} = \frac{A F}{D G}$ , so $A G \times D G = A F \times D B$ .

Step-by-step working

1
Since $D EFG$ is a square, $GF ∥ BE$
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2
$∠ GB D = ∠ A GF$ (corresponding angles)
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3
$△ G A F \sim △ B D G$ (AA similarity)
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4
$\frac{A G}{D B} = \frac{A F}{D G} \Rightarrow A G \times D G = A F \times D B$
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Concepts used

similar trianglesproveright trianglesquare

Approach

Step-by-step working

Concepts used

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