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.Question(slot="Question" question=state and prove Basic proportionality theorem" {.new row}# Basic Proportionality Theorem (BPT)

## Statement If a line is drawn parallel to one side of a triangle to intersect the other two sides, then it divides the two sides in the same ratio.

## Given

  • Triangle ( \triangle ABC )
  • Line ( DE \parallel BC ), intersecting ( AB ) at ( D ) and ( AC ) at ( E )

## To Prove [ \frac{AD}{DB} = \frac{AE}{EC} ]

## Proof

### Construction Draw ( DE \parallel BC ) intersecting ( AB ) at ( D ) and ( AC ) at ( E ).

### Proof

1. Triangles ( \triangle ADE ) and ( \triangle ABC ) have corresponding angles equal:

  • Since ( DE \parallel BC ), ( \angle ADE = \angle ABC ) (corresponding angles)
  • ( \angle AED = \angle ACB ) (corresponding angles)
  • ( \angle A = \angle A ) (common angle)

2. Using the concept of similar triangles:

  • By AA similarity criterion, ( \triangle ADE \sim \triangle ABC )
  • Therefore, their corresponding sides are in the same ratio: [ \frac{AD}{AB} = \frac{AE}{AC} \quad \text{(i)} ] [ \frac{AD}{DB} = \frac{AE}{EC} \quad \text{(ii)} ]

3. Simplifying the ratio:

  • From equation (ii): [ \frac{AD}{DB} = \frac{AE}{EC} ]

### Conclusion Hence, it is proven that if a line is drawn parallel to one side of a triangle to intersect the other two sides, then it divides the two sides in the same ratio: [ \frac{AD}{DB} = \frac{AE}{EC} ]

Hence proved.

.Question(slot="Question" question=state and prove converse of Basic proportionality theorem" {.new row}# Converse of the Basic Proportionality Theorem (CBPT)

## Statement If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

## Given

  • In ΔABC, a line DE intersects AB at D and AC at E such that (\frac{AD}{DB} = \frac{AE}{EC}).

## To Prove

  • DE || BC

## Proof

1. Start with the Given Ratios: [ \frac{AD}{DB} = \frac{AE}{EC} ]

2. Consider triangles ADE and ABC: - In ΔADE and ΔABC, DE is not necessarily parallel to BC. We need to prove DE || BC.

3. Assume DE is not parallel to BC: - Suppose DE is not parallel to BC. Draw a line DE' parallel to BC, intersecting AB at D' and AC at E'.

4. By Basic Proportionality Theorem (BPT): - Since DE' || BC, by BPT, [ \frac{AD'}{D'B} = \frac{AE'}{E'C} ]

5. But given: [ \frac{AD}{DB} = \frac{AE}{EC} ] - By assumption, DE' || BC implies: [ \frac{AD'}{D'B} = \frac{AE'}{E'C} ]

6. Since the ratios are equal: - We have: [ \frac{AD}{DB} = \frac{AD'}{D'B} \quad \text{and} \quad \frac{AE}{EC} = \frac{AE'}{E'C} ]

7. Implies: - This implies that D coincides with D' and E coincides with E'. Thus, DE and DE' are the same lines.

8. Conclusion: - Therefore, DE is parallel to BC.

Hence, we have proved that if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. [ \boxed{DE || BC} ] {.new row}Hence proved.

work done by shenith

Quiz.md grow(Width=380)

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'sin60cos30+sin30cos60=1' *Now, sin6@cos30+sin30c0s60 = [frac{sqrt3|2}][frac{sqrt3|2}]+[frac{1){2}][frac{1}{2}] (step-1) *(new-row) sin60cos30+sin30c0s60 = [frac(3)(4}]+[frac(1)(4}] (step-2) *(new-row) sin60cos30+sin30cos60 = frac{4}{4}=1 (step-3) *(new-row)we have found answer. (step-4)

id: pattern-one

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2tan^2 45 + COS^2 30 - Sin^2 60=2'

*Now, sin60cos30+sin30c0s60 = 2(1)^2 + [frac{sqrt3)(2}]^2 (step-1) *{new-row} sin60cos30+sin30cos60 = 2-frac{3}{4}+frac{3}{4} (step-2) *{new-row} sin60cos30+sin30cos60 = 2 (step-3) *{new-row} we have found answer (step-4)

id: pattern-one

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'frac{cos45} {sec30+cosec30}=[frac{3sqrt2-sqrt6} {8}]'

*Now, frac{cos45}{sec30+cosec30}=[frac{1){sqrt2}/{2}{sqrt3}+2] (Step-1) *{new-row} frac{cos45}{sec30+cosec30}=[frac{1){sqrt2}{2+2sqrt3}{sqrt3}] (Step-2) {new-row} frac{cos45}{sec30+cosec30}=[frac{1}{sqrt2}{sqrt2(2+2sqrt3)}] (Step-3) *{new-row} frac{cos45}{sec30+cosec30}=[frac{sqrt3}{2sqrt2(sqrt3+1)}] (Step-4) *{new-row} frac{cos45}{sec30+cosec30}=[frac{sqrt3}{2sqrt2(sqrt3+1)}*frac{sqrt2}(sqrt3-1){sqrt2}{sqrt3-1}] (Step-5)

*{new-row} frac{cos45}{sec30+cosec30}=[frac{3sqrt2-6}{4(3-1)}] (Step-6) *{new-row} frac{cos45}{sec30+cosec30}=[frac{3sqrt2-sqrt6}{8}] (Step-7) *{new-row} We have found

  1. Define variable A.

  2. Identify trigonometric identities:

a. sec^2 A = 1/cos^2 A

b. tanA2 A = sin^2 A / cos^2 A

  1. Substitute identities into the expression:

9(1/cos^2 A) - 9(sin^2 A / cos^2 A)

  1. Simplify the expression:

9/COSA2 A - 9(sin^2 A / COS^2 A)

= 9/cos^2 A - 9tan^2 A

  1. Factor out 9:

9(1/cos^2 A - tan^2 A)

  1. Define variable A.

  2. Identify trigonometric identities: a. cot A = 1/tan A b. tan A = 1/cot A c. sin A = cos A / cot A d. sec A = 1/sin A

  3. Express trigonometric ratios in terms of cot A:

a. cot A = 1/tan A b. tan A = 1/cot A c. sin A = cos A / cot A d. sec A = 1/(cos A / cot A) = cot A / cos A

work done by sminika

Is zero a rational number? Can you write it in the form ( \frac{p}{q} ), where ( p ) and ( q ) are integers and ( q ) is not equal to 0.

Answer: [ yes | no ]

Explanation: Zero is a rational number because it can be expressed as ( \frac{0}{1} ) or ( \frac{0}{q} ) where ( q ) is any non-zero integer.

##q2 Find six rational numbers between 3 and 4.

Possible answers:

##q3 Find five rational numbers between 3/5 and 4/5.

Possible answers:

##q4 State whether the following statements are true or false. Give reasons for your answers.

(i) Every natural number is a whole number. Answer:

Explanation: Natural numbers is a set of all positive and non zero integers. Whereas whole numbers is as set of all non negative integers inclusive of zero. Hence every natural number is a whole number too. (ii) Every integer is a whole number. Answer:

Explanation: Whole numbers include natural numbers (that begin from 1 onwards), along with 0. Integers include negative numbers, positive numbers, and zero.As the range of integers is larger than that of whole numbers, every integer is not a whole number. (iii) Every rational number is a whole number. Answer:

Explanation:A rational number can be expressed as decimal. Whole number is a positive number without a fraction or decimal. But, a rational number is any number that can be expressed as a fraction. Thus, every rational number is not a whole number.

Exercise 1.2 (not complete) ###q1 State whether the following statements are true or false. Justify your answers.

(i) Every irrational number is a real number. Answer:

Explanation: Irrational numbers are the type of real numbers that cannot be expressed in the rational form , where are integers and ( q ) ≠ 0 . In simple words, all the real numbers that are not rational numbers are irrational. (ii) Every point on the number line is of the form m , where √m is a natural number. Answer:

Explanation: : A number line may have a negative or positive number. Since, no negative can be the square root of a natural number, thus every point on the number line cannot be in the form of √m, where m is a natural number. (iii) Every real number is an irrational number. Answer:

Explanation: Real numbers consist of both irrational and rational number, every real number is not irrational.

Are the square roots of all positive integers irrational? Answer: If not, give an example of the square root of a number that is a rational number. Possible answers: