Advertisements
Advertisements
प्रश्न
Prove that the angle bisector of a triangle divides the side opposite to the angle in the ratio of the remaining sides.
उत्तर
Consider ΔABC,
Observe the following figure.
CE bisects ∠ACB.
Draw a line parallel to ray CE passing through the point B.
Extend AC so as to intersect it at D.
Line CE is parallel to line BD and AD is the transversal.
∴ ∠ACE = ∠CDB [corresponding angles] ….(1)
Now consider BC as the transversal.
∴ ∠ECB = ∠CBD [alternate angles] ….(2)
But ∠ACE = ∠ECB [given] ….(3)
∴ ∠CBD = ∠CDB [from (1), (2) and (3)
In ΔCBD, side CB = side CD [sides opposite to equal angles]
∴ CB = CD ….(4)
Now in DABD, seg EC || side BD [construction]
`therefore"AE"/"EB"="AC"/"CD"` [B.P.T] .............(5)
`therefore"AE"/"EB"="AC"/"CB"` [from equations (4) and (5)]
Thus, the angle bisector of a triangle divides the side opposite to the angle in the ratio of the remaining sides.
APPEARS IN
संबंधित प्रश्न
E and F are points on the sides PQ and PR, respectively, of a ΔPQR. For the following case, state whether EF || QR.
PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm
The given diagram shows two isosceles triangles which are similar. In the given diagram, PQ and BC are not parallel; PC = 4, AQ = 3, QB = 12, BC = 15 and AP = PQ.
Calculate:
- the length of AP,
- the ratio of the areas of triangle APQ and triangle ABC.
The two similar triangles are equal in area. Prove that the triangles are congruent.
In the given figure, ΔODC~ΔOBA, ∠BOC = 115° and ∠CDO = 700.
Find (i) ∠DCO (ii) ∠DCO (iii) ∠OAB (iv) ∠OBA.
In the given figure, ∠ABC = 90° and BD⊥AC. If BD = 8cm, AD = 4cm, find CD.
P and Q are points on the sides AB and AC respectively of a ΔABC. If AP = 2cm, PB = 4cm, AQ = 3cm and QC = 6cm, show that BC = 3PQ.
In the given figure, ∠1 = ∠2 and `(AC)/(BD)=(CB)/(CE)` Prove that Δ ACB ~ Δ DCE.
In the given figure, ∠PQR = ∠PST = 90° ,PQ = 5cm and PS = 2cm.
(i) Prove that ΔPQR ~ ΔPST.
(ii) Find Area of ΔPQR : Area of quadrilateral SRQT.
Δ ABC is similar to Δ PQR. If AB = 6cm, BC = 9cm, PQ = 9cm and PR = 10.5cm, find the lengths of AC and QR.
A triangle ABC is enlarged, about the point O as centre of enlargement, and the scale factor is 3. Find : BC, if B' C' = 15 cm.
In the given figure, PQ || AB; CQ = 4.8 cm QB = 3.6 cm and AB = 6.3 cm. Find :
- `(CP)/(PA)`
- PQ
- If AP = x, then the value of AC in terms of x.
In the given figure, ΔABC ~ ΔADE. If AE : EC = 4 : 7 and DE = 6.6 cm, find BC. If 'x' be the length of the perpendicular from A to DE, find the length of perpendicular from A to BC in terms of 'x'.
Prove that, if a line is drawn parallel to one side of a triangle to intersect the other two sides, then the two sides are divided in the same ratio.
If ΔABC, D and E are points on AB and AC. Show that DE || BC for each of the following case or not:
AB = 5.6cm, AD = 1.4cm, AC = 7.2cm, and AE = 1.8cm
In the figure, PQR is a straight line and PS || RT. If QS = 12cm, QR = 15cm, QT = 10cm and RT = 6cm, find PQ and PS.
ΔABC is enlarged, with a scale factor 5. Find: A'B', if AB = 4cm
A map is drawn to scale of 1:20000. Find: The distance covered by 6cm on the map
If ∆ABC is an isosceles triangle with ∠C = 90° and AC = 5 cm, then AB is
D is the mid point of side BC and AE ⊥ BC. If BC = a, AC = b, AB = c, ED = x, AD = p and AE = h, prove that b2 + c2 = `2"p"^2 + "a"^2/2`
ΔDEF ~ ΔABC. If DE : AB = 2 : 3 and ar ΔDEF is equal to 44 square units then ar (ΔABC) (square unit) is ______.
