in a quadrilateral ABCD, BC is parallel to AD. AB = BC, 13CD = 20BC and 34AB = 13AD. H is a point on AD such that CH is perpendicular to AD. Prove that 5CH = 3CD
Proof:
Since BC is parallel to AD, we know that the angles between AB and BC and between AD and CH are equal.
Let angle ABC = x
Then angle ADH = x
Since AB = BC, we know that angle ABC = angle BCD = x
Since 13CD = 20BC, we know that angle BCD = (20/13)x
Since 34AB = 13AD, we know that angle ABC = (13/34)x
Since CH is perpendicular to AD, we know that angle ADH = 90°
Therefore, (20/13)x + (13/34)x + 90° = 180°
Simplifying, we get 5x + 3x + 90° = 180°
Solving for x, we get x = 30°
Therefore, angle ABC = angle BCD = 30°
Since CH is perpendicular to AD, we know that angle CHD = 90°
Therefore, angle CHD + angle BCD = 90° + 30° = 120°
Since angle CHD = angle BCD, we know that 5CH = 3CD
To prove that 5CH = 3CD, we can break down the proof into steps using the given information and properties of parallel lines.
Step 1: Given information
- Quadrilateral ABCD with BC parallel to AD.
- AB = BC (1)
- 13CD = 20BC (2)
- 34AB = 13AD (3)
Step 2: Understanding the problem
- We need to prove that 5CH = 3CD.
Step 3: Proving the claim
- Since BC is parallel to AD, it implies ∠BCH = ∠CHA, and ∠CHB = ∠HCA, as corresponding angles formed by parallel lines.
- Let x be the length of AB and BC.
- Using equation (1) we can write AB = BC = x.
- Using equation (2), we can write 13CD = 20x.
- Rearranging the equation, CD = (20/13)x.
- Using equation (3), we can write 34x = 13AD.
- Rearranging the equation, AD = (34/13)x.
- Therefore, AH = AD - DH = (34/13)x - CH.
- Using the property of perpendicular lines, we know that ∠CDH = 90 degrees.
- Applying the Pythagorean theorem in right triangle CDH:
- (CH)^2 + (DH)^2 = (CD)^2
- (CH)^2 + (5CH/3)^2 = ((20/13)x)^2
- (CH)^2 + (25/9)(CH)^2 = ((20/13)^2)(x)^2
- (1 + (25/9))(CH)^2 = ((20/13)^2)(x)^2
- ((9/9) + (25/9))(CH)^2 = ((20/13)^2)(x)^2
- ((34/9))(CH)^2 = ((20/13)^2)(x)^2
- (34)(CH)^2 = ((20/13)^2)(x)^2
- (34)(CH)^2 = ((20^2)/(13^2))(x^2)
- (34)(CH)^2 = (20^2)(x^2)/(13^2)
- (CH)^2 = ((20^2)(x^2)/(13^2))/(34)
- (CH)^2 = 400x^2/(169 * 34)
- (CH)^2 = 10x^2/169
- Taking the square root of both sides, we get:
- CH = sqrt(10x^2/169)
- CH = (sqrt(10)/13)x
- Multiplying both sides by 5/3 to get 5CH:
- 5CH = (5/3)(sqrt(10)/13)x
- 5CH = (5sqrt(10)/39)x
Step 4: Comparing with 3CD
- CD = (20/13)x
- Multiplying both sides by 3:
- 3CD = (60/13)x
Step 5: Conclusion
- Comparing the expressions, we can see that 5CH does not equal 3CD.
Therefore, the claim that 5CH = 3CD cannot be proven using the given information.