Question

4. Sketch two triangles (You do not have to provide the sketch). Label the lengths of the sides of Triangle A as 3, 4, and 5. Label the lengths of the sides of Triangle B as 5, 12, and 13.
a. What is the sum of the measures of the acute angles of any right triangle? Explain your reasoning.
b. Write the tangent ratios for the acute angles of Triangle A.
c. Write the tangent ratios for the acute angles of Triangle B.
d. Write a rule describing the relationship between the tangents of the acute angles of any right triangle.

Can some one translate what a. is trying to say

Answer 1

Given a triangle with side lengths of 3, 4 and 5. This gives a right triangle with legs of length 3 and 4 and hypothenus of length 5.
Given a triangle with side lengths of 5, 12 and 13. This gives a right triangle with legs of length 5 and 12 and hypothenus of length 13.

a.) The sum of the measures of the acute angles of any right triangle is 90 degrees. This is because, for a right triagle, one of the angles is a right angle which is 90 degrees and the sum of the interior angles of a triangle is 180 degrees. Thus the other two angles of a right triangle sums up to 180 - 90 = 90 degrees.

b.) The tangent ratio of a right triangle is the ratio of the legs of the triangle. (i.e. the sides of the right triangle other than the hypothenus).
For the first right triangle with side lengths of 3, 4 and 5 units, the tangent ratios are 3/4 and 4/3.


c.) Similarly, for the second reight triangle with side lengths of 5, 12 and 13 units, the tangent ratios are 5/12 and 12/3.

d.) The rule describing the relationship between the tangents of the acute angles of any right triangle is given by
$\tan{\theta}= \frac{opposite}{adjacent}$
where: 'opposite' refers to the side opposite the angle of reference and  'adjacent' refers to the side (other than the hypothenus) adjacent the angle of reference of the right triangle.