Question

In the triangle \(\mathrm{PQR}, \mathrm{AB}\) is parallel to \(\mathrm{QR}\). The ratio of the areas of two similar triangles \(\mathrm{PAB}\) and \(\mathrm{PQR}\) is \(1: 2\). Then \(\mathrm{PQ}: \mathrm{AQ}=\) (1) \(\sqrt{2}: 1\) (2) \(1: \sqrt{2}-1\) (3) \(1:(\sqrt{2}+1)\) (4) \(\sqrt{2}: \sqrt{2}-1\)

Short answer

Answer: The ratio of lengths PQ to AQ is \(1: \sqrt{2}-1\).

Step-by-step solution

Understanding Similar Triangles Properties

When two triangles are said to be similar, this means that their corresponding angles are equal, and the ratio of their corresponding sides is constant. In our problem, Triangle PAB is similar to Triangle PQR, so we have the equality of the corresponding angles and a constant ratio for their corresponding sides.

Establishing Corresponding Ratios

Since we know that Triangle PAB is similar to Triangle PQR and line segment AB is parallel to line segment QR, we can write the ratios of corresponding sides as: - PA/PQ = AB/QR - PB/PR = AB/QR We are also given that the ratio of the areas of Triangle PAB and Triangle PQR is 1:2, which can be written as (Area of Triangle PAB)/(Area of Triangle PQR) = 1/2.

Applying Area Ratio Formula for Similar Triangles

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. That is: (Area of Triangle PAB)/(Area of Triangle PQR) = (PA/PQ)^2 Now, substitute the given area ratio 1/2: 1/2 = (PA/PQ)^2

Solving for the Ratio of Side Lengths

We now need to find the value of the ratio PA/PQ. From the area ratio equation, we have: (PA/PQ)^2 = 1/2 Taking the square root of both sides, we get: PA/PQ = 1/\sqrt{2}

Inverting the Ratio to Find PQ/AQ

We have found the ratio PA/PQ, but we need to find PQ/AQ. To do this, we simply invert the fraction: PQ/PA = \sqrt{2} Now, notice that PQ is the sum of PA and AQ, so PQ = PA + AQ. Therefore, we can substitute this into our ratio and solve for PQ/AQ: \sqrt{2} = (PA + AQ)/PA

Solving the Equation for PQ/AQ

We can now solve the equation for PQ/AQ: \sqrt{2} = (PA + AQ)/PA Multiplying both sides by PA gives: \sqrt{2} * PA = PA + AQ Subtracting PA from both sides yields: (\sqrt{2} - 1) * PA = AQ Finally, dividing both sides by PA gives the ratio PQ/AQ: PQ/AQ = 1: (\sqrt{2}-1) So, the correct answer is choice (2) \(1: \sqrt{2}-1\).

Key concepts

Properties of Similar Triangles

When we talk about the properties of similar triangles, we're looking at one of the core concepts in geometry that students must master. Similar triangles have exactly the same shape but are different in size. This congruence in shape is due to two key properties: firstly, all corresponding angles between the triangles are equal, and secondly, the ratio of the lengths of corresponding sides is consistent across both triangles. This means that if triangle ABC is similar to triangle DEF, then angle A equals angle D, angle B equals angle E, and angle C equals angle F. Moreover, the side lengths relate to each other in a proportion, so AB/DE = BC/EF = AC/DF. Understanding these properties is crucial as they form the baseline for solving numerous problems related to triangle geometry, including the exercise at hand.

For example, if a student is given that triangle XYZ is similar to triangle ABC, they can deduce that the sides of XYZ are proportional to the sides of ABC. This allows for calculations regarding lengths, heights, distances, and further application into real-world problems such as determining the height of an inaccessible object through the concept of shadow lengths.

Ratio of Corresponding Sides

The ratio of corresponding sides is a powerful tool used in solving problems involving similar triangles. It's a comparison of one side length from one triangle to the side length of the other triangle that occupies the equivalent position. When triangles are similar, their corresponding sides have lengths that are in the same proportion to each other. This ratio remains constant no matter the overall size of the triangles.

Let's delve into a practical scenario: imagine two triangles, one with side lengths of 3 cm, 4 cm, and 5 cm, and another with side lengths of 6 cm, 8 cm, and 10 cm. These triangles are similar because their side lengths are in a 1:2 ratio (3/6 = 4/8 = 5/10). If we only knew the side lengths of the smaller triangle and the ratio, we could easily find the corresponding side lengths of the larger triangle. This concept can also be extended to other applications such as scale models, where a smaller or larger version of an object retains the same proportions as the original.

Area Ratio Formula in Triangles

The area ratio formula for triangles provides a fantastic shortcut for comparing the sizes of similar triangles without the need for tedious calculations. The formula states that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Mathematically presented, if triangle ABC is similar to triangle DEF, then the area of triangle ABC divided by the area of triangle DEF equals (AB/DE) squared.

This formula is immensely helpful in various mathematical problems, including the example in the textbook solution. When the problem states that the ratio of the areas of two similar triangles PAB and PQR is 1:2, we can square the ratio of the corresponding sides to find this area ratio. If PA/PQ has a ratio of x, then the area of PAB to the area of PQR is x squared. Knowing this formula allows students to work backwards too; given an area ratio, they can deduce the side lengths ratio. This concept is not only crucial for answering textbook problems but also in real-life applications such as land surveying and architecture.