Question

In triangle abc, ∠c is a right angle and cd is the altitude to ab . find the angles in δcbd and δcad if: b m(∠a) = a.

A) arctan∠CDB = 90∘ −arctan(b), arctan ∠CBD = arctan(b), ∠CAD = arctan(b)
B) arctan∠CDB = arctan(b), arctan ∠CBD = 90∘-arctan(b), ∠CAD = arctan(b)
C) arctan ∠CDB = arctan(b), arctan ∠CBD =90∘ −arctan(b), arctan ∠CAD = arctan(b)
D) arctan∠CDB = 90∘ −arctan(b), arctan ∠CBD = arctan(b), arctan∠CAD = arctan(b)

Answer 1

The measures of the angles in the triangles ΔCBD and ΔCAD are found using trigonometric relationships. ∣CDB is 90° - arctan(b) and both ∣CBD and ∣CAD are arctan(b), based on the given that m(∣A) = arctan(b) and the properties of right triangles and altitudes.

Step-by-step explanation:

The question is about finding the angles in the triangles ΔCBD and ΔCAD given that ∣C is a right angle and CD is the altitude to AB. Additionally, it is known that m(∣A) = arctan(b). Understanding that the altitude from the right angle in a right triangle creates two smaller right triangles, we can use trigonometric relationships to find the measures.

Since ∣C is a right angle, ∣ACD and ∣BCD are complementary in ΔACD and ΔCBD respectively; thus, ∣ACD plus ∣BCD equals 90°.

Given that the measure of ∣A is arctan(b), ∣ACD will also be arctan(b) since angle CAD and angle A are the same because CD is the altitude and creates two congruent angles.

∣CDB, being the complementary angle to ∣ACD in triangle ΔCDB, will have a measure of 90° - arctan(b), since in a right triangle the sum of the acute angles equals 90°.

Therefore, ∣CBD, being opposite to side AB, will have a measure of arctan(b) because it is equal to ∣A due to the altitude creating similar triangles.