Problem 78 - Elementary Algebra

Question

The perimeter of an isosceles triangle \(A B C\) measures 11 units and its two equal sides measure 4 units. If triangle \(A B C\) is similar to triangle RST and triangle RST has a perimeter of 22 units, then find all the sides of triangle RST.

Step-by-Step Solution

Answer

The sides of triangle RST are 8 units, 8 units, and 6 units.

Step 1: Understand the Similarity Ratio

The perimeter of triangle ABC is given as 11 units. Both the equal sides measure 4 units each. The perimeter of triangle RST is given as 22 units. Since the triangles are similar, their side lengths are proportional. The similarity ratio between the triangles is based on their perimeters, so the ratio is \( \frac{22}{11} = 2 \).

Step 2: Calculate the Third Side of Triangle ABC

The perimeter of triangle ABC is the sum of all its sides, meaning \(4 + 4 + x = 11\). Solving for \(x\) gives \(x = 11 - 8 = 3\). The side lengths of triangle ABC are 4, 4, and 3 units.

Step 3: Apply the Similarity Ratio to Triangle RST

Using the similarity ratio of 2, scale the sides of triangle ABC to find the sides of triangle RST. Each side length of triangle ABC is multiplied by the ratio 2. Calculate the side length corresponding to the 4-unit sides in ABC: \(4 \times 2 = 8\). For the 3-unit side: \(3 \times 2 = 6\).

Step 4: Verify the Perimeter of Triangle RST

The sides of triangle RST are now 8, 8, and 6. Verify this against the given perimeter of triangle RST: \(8 + 8 + 6 = 22\). This confirms the dimensions fit the condition of the problem.

Key Concepts

Similar Triangles

When two triangles are similar, it means they have the same shape but different sizes. The angles of the triangles are congruent (equal in measure), and their corresponding sides are in proportion. This concept is useful for solving problems involving triangles because if you know the sides of one triangle, you can determine the other by using a similarity ratio.

• For triangles to be similar, their corresponding angles must be equal.
• The ratio of any two corresponding lengths in the triangles is consistent.

In the problem, triangles ABC and RST are similar, meaning their corresponding side lengths are proportional. You can find the lengths of unknown sides using the similarity ratio derived from known sides.

Perimeter

The perimeter of a triangle is the total length around the triangle. You calculate it by adding up all the side lengths of the triangle.

• Perimeter for a triangle with sides a, b, and c: \( P = a + b + c \).
• Triangles with the same perimeter can differ in side length proportions.

For triangle ABC, the perimeter is given as 11 units, while for triangle RST, it's 22 units. These perimeters help establish the similarity ratio, crucial for solving the problem. Knowing how perimeter connects to side lengths is key in determining unknown measures.

Isosceles Triangle

An isosceles triangle has two sides that are equal in length, and consequently, the angles opposite these sides are also equal. Properties specific to isosceles triangles can simplify problem-solving, such as quickly identifying the structure and relationships between sides and angles.

• Common form: two equal sides \( a \), base side \( b \).
• Identifying isosceles triangles can aid in verifying side lengths.

In this problem, triangle ABC is an isosceles triangle with two sides measuring 4 units each. Recognizing its isosceles nature helps efficiently calculate the missing side and apply those findings to its similar triangle, RST.

Proportional Reasoning

Proportional reasoning involves understanding and working with ratios and proportions. It is incredibly helpful when dealing with similar triangles, as it allows you to scale lengths and other measurements based on a known ratio.

• A ratio compares two quantities, such as \( \frac{a}{b} \).
• Proportional reasoning helps in scaling figures and solving for unknowns.

In this scenario, the ratio of the perimeters (\( \frac{22}{11} = 2 \)) gives the similarity ratio. This ratio indicates that each side of triangle RST is twice that of triangle ABC. By applying this ratio, you can calculate the sides of triangle RST by multiplying the known sides of triangle ABC by the ratio.