Question

In figure 2.21, 2 DFE = 90°.
FG LED, IF GD = 8, FG = 12,
find (1) EG (2) FD and (3) E​

Answer 1

Correct Question:

In figure, ∠DFE = 90°, FG ⊥ ED. If GD = 8, FG = 12, find 1) EG 2) FD and 3) EF.

Answer:

1) EG = 18 units

2) FD = 4 √13 units

3) EF = 6 √13 units

Step-by-step-explanation:

NOTE: Refer to the attachment for the diagram.

In figure,

We have given that,

∠DFE = 90°

FG ⊥ ED

GD = 8

FG = 12

1)

In △DFE,

∠DFE = 90° - - [ Given ]

Seg FG ⊥ hypotenuse ED - - [ Given ]

∴ FG² = DG × EG - - [ Property of geometric mean ]

⇒ ( 12 )² = 8 × EG

⇒ EG = 12 × 12 ÷ 8

⇒ EG = 12 × 3 ÷ 2

⇒ EG = 6 × 3

⇒ EG = 18 units

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2)

In △DGF,

∠DGF = 90° - - [ FG ⊥ ED ]

∴ FD² = DG² + GF² - - [ Pythagors theorem ]

⇒ FD² = 8² + ( 12 )²

⇒ FD² = 64 + 144

⇒ FD² = 208

⇒ FD = √(16 × 13) - - [ Taking square roots ]

⇒ FD = 4 √13 units

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3)

In △EGF,

∠EGF = 90° - - [ Given ]

∴ EF² = EG² + GF² - - [ Pythagors theorem ]

⇒ EF² = ( 18 )² + ( 12 )²

⇒ EF² = 324 + 144

⇒ EF² = 468

⇒ EF = √(36 × 13) - - [ Taking square roots ]

⇒ EF = 6 √13 units

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Additional Information:

1. Geometric mean theorem:

1. The perpendicular segment from the vertex opposite to hypotenuse in a right angled triangle is called the geometric mean of the triangle.

2. It divides the hypotenuse in two parts.

3. The square of the segment is equal to the product of the two parts in which the hypotenuse is divided.