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Apollonius's Theorem

Assertion AIn...

Question

Assertion (A)
In ∆ABC, if DE ∥ BC intersects AB in D and AC in E, then $\frac{A D}{A B} = \frac{A E}{A C} .$
Reason (R)
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then these sides are divided in the same ratio.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A)
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

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Solution

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

The Reason (R) is clearly true by Thales' theorem.
It is given that DE $∥$ BC.
Applying Thales' theorem, we have:

$\frac{A D}{D B} = \frac{A E}{E C} \Rightarrow \frac{D B}{A D} = \frac{E C}{A E} \Rightarrow 1 + \frac{D B}{A D} = 1 + \frac{E C}{A E} \Rightarrow \frac{A B}{A D} = \frac{A C}{A E} \Rightarrow \frac{A D}{A B} = \frac{A E}{A C} Therefore, Assertion (A) is true and (R) gives (A) .$

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Similar questions

Q. Assertion (A)
In an isosceles ∆ABC, if ∠C = 90°, then AB² = 3AC².
Reason (R)
In an isosceles ∆ABC, if AC = BC and AB² = 2AC², then ∠C = 90°.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A)
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

Q. Assertion (A)
∆ABC ∼ ∆DEF, such that ar(∆ABC) = 100 cm² and ar(∆DEF) = 144 cm². If AB = 24 cm, then DE = 36 cm.
Reason (R)
If ∆ABC ∼ ∆DEF, then

\frac{ar (∆ A B C)}{ar (∆ D E F)} = \frac{A B^{2}}{D E^{2}} = \frac{B C^{2}}{E F^{2}} = \frac{A C^{2}}{D F^{2}} .

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

Q. Assertion (A)
∆ABC ∼ ∆DEF such that BC = 4.5 cm and EF = 3.5 cm and ar(∆ABC) = 81 cm², then ar(∆DEF) = 49 cm².
Reason (R)
The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding medians.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A)
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

Q. Assertion (A)
If on dividing the polynomial p(x) = x² − 3ax + 3a − 7 by (x + 1), we get 6 as remainder, then a= 3.

Reason (R)
When a polynomial p(x) is divided by (x − α), then the remainder is p(α).

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
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Q. Assertion (A)
If the volumes of two spheres are in the ratio 27 : 8, then their surface areas are in the ratio 3 : 2.

Reason (R)
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\frac{4}{3} π R^{3}

Surface area of a sphere =

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(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
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