Advertisements
Advertisements
प्रश्न
□ABCD is a parallelogram. Point P is the midpoint of side CD. seg BP intersects diagonal AC at point X, then prove that: 3AX = 2AC
उत्तर
Proof: From the figure, in ΔABX and ΔCPX
As, AB || CD
∠BAX = ∠PCX ......[Alternate angle]
∠BXA = ∠PXC ........[Vertically opposite angles]
∴ ΔABX ∼ ΔCPX .......[By AA similarity theorem]
We know that,
Similar triangles have comparable side ratios that are similar to or equal.
∴ `("AX")/("CX") = ("AB")/("CP")`
But CD = AB and 'P' is mid-point of CD.
∴ AB = 2CP
∴ `("AX")/(("AC" - "AX")) = (2"CP")/("CP")`
∴ `("AX")/(("AC" - "AX"))` = 2
∴ AX = 2(AC – AX)
∴ AX = 2AC – 2AX
∴ AX + 2AX = 2AC
∴ 3AX = 2AC
Hence proved.
APPEARS IN
संबंधित प्रश्न
State which pair of triangles in the following figure are similar. Write the similarity criterion used by you for answering the question, and also write the pairs of similar triangles in the symbolic form:
State which pair of triangles in the following figure are similar. Write the similarity criterion used by you for answering the question, and also write the pairs of similar triangles in the symbolic form:
A vertical pole of a length 6 m casts a shadow 4m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.
In the following Figure, DE || BC such that AE = (1/4) AC. If AB = 6 cm, find AD.
In the following figure, PA, QB and RC are each perpendicular to AC. Prove that `1/x+1/z=1/y`
ABCD is a parallelogram and APQ is a straight line meeting BC at P and DC produced at Q. Prove that the rectangle obtained by BP and DQ is equal to the AB and BC.
The sides of certain triangles are given below. Determine which of them right triangles are.
1.4cm, 4.8cm, 5cm
In the given figure, DE║BC such that AD = x cm, DB = (3x + 4) cm, AE = (x + 3) cm and EC = (3x + 19) cm. Find the value of x.
In a trapezium ABCD, it is given that AB║CD and AB = 2CD. Its diagonals AC and BD intersect at the point O such that ar(ΔAOB) = 84cm2. Find ar(ΔCOD).
The corresponding sides of two similar triangles are in the ratio 2 : 3. If the area of the smaller triangle is 48 cm2, find the area of the larger triangle.
Two triangles DEF an GHK are such that ∠D = 48° and ∠H = 57° . If ΔDEF ∼GHK then find the measures of ∠F
In the given figure, value of x(in cm) is
In figure, if AB || DC and AC and PQ intersect each other at the point O, prove that OA . CQ = OC . AP
In figure, ABC is a triangle right angled at B and BD ⊥ AC. If AD = 4 cm, and CD = 5 cm, find BD and AB.
In triangles ABC and DEF, ∠B = ∠E, ∠F = ∠C and AB = 3DE. Then, the two triangles are ______.
In the figure with ΔABC, P, Q, R are the mid-points of AB, AC and BC respectively. Then prove that the four triangles formed are congruent to each other.
In the given figure, ∠ADC = ∠BCA; prove that ΔACB ∼ ΔADC. Hence find BD if AC = 8 cm and AD = 3 cm.
In ΔABC, DE || AB. If AB = a, DE = x, BE = b and EC = c. Then x expressed in terms of a, b and c is ______.
If ΔABC ∼ ΔDEF such that ∠A = 92° and ∠B = 40°, then ∠F = ?
