In a triangle ABC, P and Q are two points on AB and AC, respectively, such that PQ is parallel to BC. If AC = 5QC, then the ratio PQ : BC is equal to:
4 : 5
To solve this problem, we use the Basic Proportionality Theorem, which states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those two sides in the same ratio.
Given that PQ is parallel to BC, and AC = 5QC, by the Basic Proportionality Theorem, we have:
\[\frac{AP}{PB} = \frac{AQ}{QC}\]
Let's denote AC as the full length, and let QC = x. Then AC = 5QC implies AC = 5x. Because Q divides AC in such a way that AQ = AC - QC, then AQ = 5x - x = 4x.
Substituting back, we have:
\[\frac{AQ}{QC} = \frac{4x}{x} = 4\]
By the Basic Proportionality Theorem, this is equal to the ratio of \( \frac{PQ}{BC} \):
\[\frac{PQ}{BC} = \frac{4}{5}\]
Thus, the ratio of PQ : BC is 4 : 5.
Important Questions from Ratio and proportion
A’s marks in Mathematics are directly proportional to practice time. In 6 hours of practice, A gets 70 marks. What should be the practice time (approximately) to get 90 marks?
The average of the areas of 2 similar triangles is 706.5 m2 whose perimeters are in the ratio of 6 : 11. What is 20% of the difference (in m2) in areas of both triangles?
In a triangle ABC, D and E are two points on sides AB and AC, respectively, such that DE is parallel to BC and AD : DB = 3 : 5. If AC = 5.6 cm, then find the value (in cm) of AE.
Find the mean proportional between 25 and 81.
8 apples and 10 oranges weigh 5 kg. 12 apples and 20 oranges weigh 9 kg. What is the weight (in kg, rounded off to the nearest integer) of 15 apples and 24 oranges?
