Question

When designing a truss, a truss builder might know the base angle measurement and the length of the tie beam needed. The next step is to compute the height of the king post. Let's take a look at some right triangles to see whether knowing the measure of an acute angle of a right triangle and the length of one of the sides is enough to find the lengths of the other two sides.

Part A: What is the measure of ∠BAC?
Part B: How are ΔABC and ΔADE related? How do you know? Explain.
Part C: Record the lengths of the sides of ΔABC and ΔADE.
Part D: Calculate the ratios in the table using the side lengths that you recorded in Part C.

Answer 1

To determine unknown sides of a right triangle, we can use trigonometric ratios and the Pythagorean theorem. The measure of angle BAC is needed, and triangles ABC and ADE are related by similarity if they share the same angles. With side lengths known, ratios like sine, cosine, and tangent are calculated to find missing measurements.

Step-by-step explanation:

Understanding Trigonometric Relationships in Right Triangles

When we examine right triangles, if we know the measure of an acute angle and the length of one side, we can use trigonometric ratios and the Pythagorean theorem to find the lengths of the other two sides.

Part A: Measure of ∠BAC

The measure of ∠BAC is the acute angle given or that can be measured in the right triangle.

Part B: Relationship Between ΔABC and ΔADE

Triangles ΔABC and ΔADE are likely to be similar if they share the same angle measures, meaning that their corresponding sides are proportionate.

Part C: Lengths of Sides

To record the lengths of the sides of the triangles, we measure or use given values, ensuring we label each side in relation to the triangle it belongs to.

Part D: Calculating Ratios

With the side lengths, we can calculate the trigonometric ratios of the sides such as sine, cosine, and tangent, which are the ratios of the opposite side to the hypotenuse, adjacent side to the hypotenuse, and opposite side to the adjacent side, respectively.