Steph B.

Secondary Math Teacher

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Geometry

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Question:

You can use indirect measurement to estimate the height of a building. You are 5 feet tall. You measure your distance from the base of the building (7 ft) and the distance from the ground to a point on the building that you are looking at (10 ft). This relationship will form triangle ABC. Maintaining the same angle of sight, move back until the top of the building is in your line of sight. You are now 42 feet from the base of the building. This relationship will form triangle XYZ. Explain why triangles ABC and XYZ are similar. Then, find the height of the building.

Steph B.

Answer:

Because both triangle share angle B, it is is congruent to itself. Angle A and angle D have the same line of sight, and so they are congruent. Because two angles are congruent, the third angles, by the Triangle Sum Theorem, are congruent. Therefore, if both triangles have the same angle measures, they are similar, by definition. As such, we can then write a proportion using the ratios of height to base to find the height of the building: x/5 = 42/7. Using the Cross Products Property, we can then find the value of x: x = 30 feet. However, we must take into account our height (or our height of sight) and add it to the value we found for x. So, 30 + 5 = 35 feet. The height of the building is 35 feet.

Pre-Algebra

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Question:

A ladder leaning against a wall forms a right triangle with angle measures x and 5x degrees. The triangle forms exterior angles with the wall and the ground. What are the measures of the exterior angles? Justify your answer.

Steph B.

Answer:

Using the Triangle Sum Theorem, we know that the interior angles of a triangle sum to be 180 degrees. Therefore, we can write the following equation: x + 5x + 90 = 180. From our equation, we can find the value of x, x = 15. Now that we know the value of x, we know that the interior angles of the triangle are 15 degrees, 90 degrees, and 75 degrees. Using the Exterior Angle Theorem (Triangles), we know that an exterior angle of a triangle is equal to the sum of the two nonadjacent interior angles. So the two exterior angles that are formed with the wall and with the ground are 15 + 90 = 105 degrees and 75 + 90 = 165 degrees.

Algebra

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Question:

Suppose {a^x}/{a^y}=a^3 and {a^x}/{a^3y}=a^-5 and . Find the values of x and y. Explain how you found your answer.

Steph B.

Answer:

Using our knowledge of the Laws of Exponents, we can display what information is given using an equation. In other words, according to the Quotient of Powers rule, x - y = 3 and x - 3y = -5. Using elimination, we can then find the values of x and y, with solutions x = 7 and y = 4. This is much more efficient than using trial and error. It gives a systematic way of approaching any problem of this type.

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