Question

\(\frac{\text { In the figure above (not to scale), } A B C D} \text { is an isosceles trapezium. }\) \(\overline{A B}|| \overrightarrow{C D}, A B=9 \mathrm{~cm}\) and \(C D=12 \mathrm{~cm} . A P: P D=B Q: Q C=1: 2\) Find \(\mathrm{PQ}\). (1) \(11 \mathrm{~cm}\) (2) \(10.5 \mathrm{~cm}\) (3) \(10 \mathrm{~cm}\) (4) \(9.5 \mathrm{~cm}\)

Short answer

Question: In an isosceles trapezium ABCD, AB || CD and AB = 9 cm, CD = 12 cm. If AP:PD = BQ:QC = 1:2, find the length of PQ. Answer: The length of PQ is 3 cm.

Step-by-step solution

Find Lengths of AP, PD, BQ, and QC

We know that AP:PD = 1:2 and BQ:QC = 1:2. We can use these ratios to find the lengths of each segment. Since AP + PD = AB = 9 cm and AB = 1x + 2x = 3x, we can find the value of x and the lengths of these segments: 3x = 9 cm => x = 3 cm So, AP = 3 cm and PD = 6 cm. Similarly, for BQ and QC, we get: 3x = CD - AB = 12 cm - 9 cm = 3 cm => x = 1 cm So, BQ = 1 cm and QC = 2 cm.

Find the height of the trapezium

Since ABCD is an isosceles trapezium, the height or altitude (let's call it h) will be the same for both triangles ABP and CDQ. Because triangle APD is a right triangle, with AD as the hypotenuse and sides PD and h, we can use the Pythagorean theorem to find the height: AD² = PD² + h² We also have another right triangle, CQD. Therefore, we can write: AD² = (QC + BQ + PD)² + h² We can now solve for h: (3+1+6)² = 10² = 100 h² = 100 - PD² = 100 - 6² = 64 h = √64 = 8 cm

Find the length of PQ

Finally, we have the height and we know that triangle ABP and triangle CDQ are similar, so their corresponding sides are proportional. Now, we can use the proportionality of these sides to find the length of PQ: (PQ + 3) / (3 + 6) = PQ / (2 + 1) Solving for PQ, we get: (PQ + 3) / 9 = PQ / 3 3(PQ + 3) = 9PQ PQ + 3 = 3PQ 2PQ = 3 PQ = 1.5 cm The length of PQ is 1.5 cm, which is not among the given options. Check the calculation again, we made a mistake in calculating the proportionality factor for BQ and QC. The correct factor is x=3. Recalculating BQ and QC: BQ = 3 cm and QC = 6 cm. Now we can correct the proportionality equation: (PQ + 3) / (3 + 6) = PQ / (6 + 3) (PQ + 3) / 9 = PQ / 9 So, PQ = 3 cm. However, this answer is not among the given options either. There might be a mistake in the problem statement or the given options themselves.

Key concepts

Proportionality Theorem

The Proportionality Theorem plays a pivotal role in solving geometric problems, especially those that involve similar triangles. This theorem states that if a line is parallel to one side of a triangle and intersects the other two sides, it divides those two sides proportionally. In the case of an isosceles trapezium like ABCD, we can use this theorem to find relationships between different segments. For instance, in the original problem, AP:PD is given as 1:2.
This ratio indicates that AP is one-third the length when AP and PD are combined, which helps in determining the exact lengths of these segments using their total length.

Understanding how to apply the Proportionality Theorem means recognizing these parallel lines and their effect on side lengths. In our problem:

• The line AB is parallel to CD, suggesting proportionality between segments AP, PD, BQ, and QC.
• You can easily determine unknown lengths by setting up equations based on given ratios and solving them.

In summary, remember that parallel lines dividing a triangle’s sides create proportional segments. This insight is crucial for problems involving parallel lines in geometry.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that relates the sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In the context of the given trapezium problem, this theorem is instrumental in finding the altitude, or height, of the trapezium. We break down the trapezium into right triangles to apply this theorem effectively.

Let's dissect the steps:

• Identify the right triangles: In our trapezium ABCD, triangles like APD or CQD can be isolated as right triangles for measurement.
• Use side lengths: For triangle APD, AD is the hypotenuse, and PD and the height are the other two sides.

By employing the Pythagorean Theorem, we calculated the height (h) of the trapezium as follows:\[ h^2 = AD^2 - PD^2 \]
Taking the known side PD as part of the equation helped solve for h, offering insights into the trapezium’s geometry. Understanding and applying the Pythagorean Theorem enables you to tackle diverse geometric challenges in a systematic manner.

Solving Linear Equations

Solving linear equations is a crucial skill in mathematics that aids in determining unknown values in an algebraic expression. In geometry problems, we often represent length ratios as linear equations, like the ones found in trapezium partitioning.

In solving the original problem, here are the crucial steps:

• Set up the equation: We know that sums of segments equal specific lengths (e.g., AP + PD = 9 cm and BQ + QC = 3 cm in the problem).
• Express unknowns: Use the given ratios, like 1:2, to express parts of these segments using a variable, which becomes our x.

From these expressions, linear equations emerge, such as:\[ 3x = 9 \, \text{to solve for } x \]\By solving these equations, you can unravel the puzzle of segment lengths, providing a clearer understanding of the geometrical figures involved.

This skill not only aids in solving current problems but also enhances your ability to handle more complex algebraic equations in broader contexts. Understanding how to construct and solve these equations is vital for any mathematical challenge that involves algebra and geometry.