5.2 Pythagorean Theorem and Similar Triangles

Right Triangles and the Pythagorean Theorem

Right triangles are a cornerstone of geometry on the SHSAT. A right triangle is a triangle that contains one interior angle measuring exactly 90° (a right angle). The two sides that form the right angle are called the legs, and the side opposite the right angle is the hypotenuse. The hypotenuse is always the longest side of a right triangle.

1. The Pythagorean Theorem

The relationship between the sides of a right triangle is defined by the Pythagorean Theorem:

$$a^2 + b^2 = c^2$$

where a and b represent the lengths of the legs, and c represents the length of the hypotenuse.

Using Pythagorean Triples to Save Time

On the SHSAT, speed is critical. While you can always use a² + b² = c² to calculate a missing side, many problems use Pythagorean triples—sets of three integers that perfectly satisfy the theorem. Recognizing these triples and their multiples allows you to find missing side lengths instantly without writing down equations.

The table below outlines the most common triples tested on the exam:

Example: If a right triangle has a leg of length 15 and a hypotenuse of length 25, you could solve 15² + b² = 25². However, if you recognize that 15 and 25 are multiples of the (3, 4, 5) triple (scaled by a factor of 5), you can immediately determine that the missing leg is 4 * 5 = 20.

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2. Special Right Triangles

There are two types of right triangles that have fixed side ratios based on their angle measures. These are called special right triangles:

The 45°-45°-90° Right Triangle (Isosceles Right Triangle)

In a triangle with angles measuring 45°, 45°, and 90°, the two legs are congruent (equal in length). The hypotenuse is always √2 times the length of a leg:

• Legs = x
• Hypotenuse = x√2

Example: If an isosceles right triangle has legs of 5 cm, the hypotenuse is 5√2 cm. If the hypotenuse is 10 cm, the legs are 10 / √2 = 5√2 cm.

The 30°-60°-90° Right Triangle

In a triangle with angles measuring 30°, 60°, and 90°, the sides are in the ratio 1 : √3 : 2:

• Side opposite 30° (short leg) = x
• Side opposite 60° (long leg) = x√3
• Side opposite 90° (hypotenuse) = 2x

Example: If the short leg is 6 inches, the hypotenuse is 12 inches, and the long leg is 6√3 inches.

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3. Similarity in Triangles

Two triangles are similar if they have the same shape but not necessarily the same size. Formally, two triangles are similar if:

1. Their corresponding angles are equal.
2. Their corresponding sides are proportional.

Similarity is denoted by the symbol ~. For example, ΔABC ~ ΔDEF means that vertex A corresponds to D, B corresponds to E, and C corresponds to F.

Proportional Sides

When two triangles are similar, you can set up equivalent ratios (proportions) to find missing side lengths. For example, if ΔABC ~ ΔDEF:

$$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$$

Standard Shadow Problems

A classic SHSAT application of similar triangles is the "shadow problem," where a light source (like the Sun or a lamppost) casts shadows of objects, creating similar right triangles.

Worked Example: A 6-foot-tall student stands next to a flagpole. At the same time of day, the student casts a shadow that is 4 feet long, and the flagpole casts a shadow that is 18 feet long. How tall is the flagpole?

Solution: Because the angle of the sun is the same for both the student and the flagpole, both objects form right angles with the ground, creating similar triangles.

• Set up a proportion comparing height to shadow length: (Height of flagpole) / (Shadow of flagpole) = (Height of student) / (Shadow of student)
• Let x be the height of the flagpole: x / 18 = 6 / 4
• Cross-multiply to solve: 4x = 18 * 6 => 4x = 108 => x = 27 The flagpole is 27 feet tall.

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4. The Scale Factor and Area Relationships

When scaling geometric figures, it is essential to understand how the scale factor affects perimeter and area.

If two similar figures have corresponding side lengths in the ratio of a:b, then:

• Ratio of Perimeters: The perimeters are in the same ratio as the side lengths, which is a:b.
• Ratio of Areas: The areas are in the ratio of the squares of the side lengths, which is a²:b².

The table below illustrates this relationship:

Worked Example: Two similar triangles, Δ1 and Δ2, have perimeters in a ratio of 3:4. If the area of the smaller triangle (Δ1) is 45 square inches, what is the area of the larger triangle (Δ2)?

Solution:

• The ratio of the perimeters is 3:4. Because perimeters scale linearly, the scale factor between the sides is also 3:4.
• The ratio of their areas must be the square of the scale factor: Area Ratio = (3/4)² = 9/16
• Set up a proportion using the known area: 9/16 = 45 / Area_2
• Cross-multiply to solve: 9 * Area_2 = 16 * 45 => 9 * Area_2 = 720 => Area_2 = 80 The area of the larger triangle is 80 square inches.

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5. Identifying Hidden Right Triangles

On the SHSAT, right triangles are often hidden inside other shapes, such as rectangles, squares, trapezoids, or isosceles triangles.

• Isosceles Triangles: Drawing an altitude (a line from the top vertex perpendicular to the base) splits the isosceles triangle into two congruent right triangles. The altitude bisects the base.
• Trapezoids: Drawing a vertical height line from one of the upper vertices to the base isolates a right triangle on the side, allowing you to use the Pythagorean Theorem to find missing side lengths or the height.