Advertisements
Advertisements
Question
In the figure, if ∠FEG ≡ ∠1 then, prove that DG2 = DE.DF
Solution
Proof:
| S.No. | Statements | Reasons |
| 1. |
∠FEG ≡ ∠1 ⇒ ∠DEG = 180° − ∠FE |
Given linear pair |
| 2. |
∠FDG + ∠DGF = 1 ∠EDG + ∠DFG = 1 |
Exterior angle = Sum of interior opposite angles ∵ ∠FDG = ∠EDG |
| 3. |
∠DEG = 180° − ∠FEG ∠DEG = 180° − ∠1 |
By 1 |
| 4. |
In ΔDFG ∠DEG = 180° − [∠FDG + ∠DFG] |
Angle sum property |
| 5. | ∠DGF = 180° − ∠1 | By 2 |
| 6. |
∠DGF = ∠DEG |
By 3 |
| 7. |
∠EDG = ∠EDG |
Common in ΔFDG and ΔEDG |
| 8. | ∴ ∠DGE = ∠DFG | Remaining angle by sum property and by 6 |
| 9. | ∴ ΔDGF ∼ ΔDEG |
By 6, 7, 8 By AAA similarity |
| 10. | `"DG"/"DE" = "GF"/"EG" = "DF"/"DG"` | Corresponding sides of similar traingle are proportional |
| 11. |
`"DG"/"DE" = "DF"/"DG"` DG.DG = DF.DE DG2 = DE.DF |
From 9 |
APPEARS IN
RELATED QUESTIONS
In ∆ABC, AP ⊥ BC, BQ ⊥ AC B– P–C, A–Q – C then prove that, ∆CPA ~ ∆CQB. If AP = 7, BQ = 8, BC = 12 then Find AC.
Δ ABC ~ Δ DEF. If BC = 3cm , EF=4cm and area of Δ ABC = 54 cm2 , find area of Δ DEF.
AD and BC are two straight lines intersecting at 0. CD and BA are perpendirulars from Band Con AD. If AB=6cm, CD =9cm, AD =20cm and BC=25cm, find the lengths of AO, BO, CO and DO.
In fig.DE || BC ,AD = 1 cm and BD = 2 cm. what is the ratio of the ar(ΔABC) to the ar (ΔADE)?
On a map drawn to scale of 1 : 2,50,000 a rectangular plot of land ABCD has the following measurement AB = 12 cm, BC = 16 cm angles A, B, C, and D are 900 each. Calculate:
(i) The diagonal distance of the plot of land in
(ii) Actual length of diagonal.
If ΔABC, D and E are points on AB and AC. Show that DE || BC for each of the following case or not:
AD = 5.7cm, BD = 9.5cm, AE = 3.3cm, and EC = 5.5cm
In a quadrilateral PQRS, the diagonals PR and QS intersect each other at the point T. If PT:TR = QT :TS = 1:2, show that TP:TQ = TR:TS
Find the scale factor in each of the following and state the type of size transformation:
Model volume = 200cm3, Actual volume = 8cm3
ΔABC and ΔBDE are two equilateral triangles such that D is the mid point of BC. Ratio of the areas of triangle ΔABC and ΔBDE is ______.
In ΔABC, DE || BC (as shown in the figure), If AD = 4 cm, AB = 9 cm and AC = 13.5 cm, then the length of EC is ______.
